1 


LIBRARY 

OF  THE 

University  of  California. 

GIFT    OF 


..sJ..,...SfV^..v...V'?^.P^^ 

Class 


7^^  .J^^  X-C^-^^^J-*-  -^ 


TREATISE 


ON 


AL  GEBE  A 


BY 


Profs.  OLIVER,  WAIT  and  JONES 


CORNELL   UNIVERSITY. 


SECOND    EDITION. 


ITHACA,   N.Y.: 
DUDLEY   F.   FINCH. 

1887. 


0  3^ 


Entered  according  to  Act  of  Congress,  In  the  year  1882,  by 
JAMES  E.  OLIVER,  LUCIEN  A.  WAIT  and  GEORGE  W.  JONES, 

in  the  oflace  of  the  Librarian  of  Congress,  at  Washington. 


^^°^ 


^v 


J.  S.  CusHiNG  &  Co.,  Printers,  Boston. 


PEEFAOE. 


In  writing  this  treatise  on  algebra,  the  authors  have  had  two 
rules  for  their  guidance. 

As  to  matter  :  '  'Assume  no  previous  knowledge  of  algebra, 
but  lay  down  the  primary  definitions  and  axioms,  and,  building 
on  these,  develop  the  elementary  principles  in  logical  order; 
add  such  simple  illustrations  as  shall  make  familiar  these  prin- 
ciples and  their  uses." 

As  to  form  :  "'  Make  clear  and  precise  definition  of  every  word 
and  symbol  used  in  a  technical  sense ;  make  formal  statement 
of  every  general  principle,  and,  if  not  an  axiom,  prove  it  rigor- 
ously ;  make  formal  statement  of  every  general  problem,  and 
give  a  rule  for  its  solution,  with  reasons,  examples,  and  checks ; 
add  such  notes  as  shall  indicate  motives,  point  out  best  arrange- 
ments, make  clear  special  cases,  and  suggest  extensions  and 
new  uses." 

In  working  out  the  plan  here  outlined,  wide  departures  have 
been  made  from  the  standard  text-books.  Many  new  things 
have  been  introduced,  not,  indeed,  because  they  were  new,  but 
necessarily,  either  as  definitions  in  giving  larger  meanings  to  old 
words,  or  as  axioms  and  theorems  in  stating  and  proving  the 
elementary  principles,  or  as  problems  and  notes  in  showing  new 
uses  of  principles  already  proved  :  e.g.^  many  fundamental  prin- 
ciples were  found  to  be  omitted  by  elementary  writers  because 
too  difficult  for  a  beginner,  and  by  subsequent  writers  as  already 
known.  A  typical  case  is  that  of  logarithms  :  that  "the  product 
of  two  powers  of  any  same  base  is  a  power  of  that  base  whose 
exponent  is  the  sum  of  the  exponents  of  the  factors  "  is  gener- 
ally proved  for  commensurable  powers,  but  assumed,  without 
proof,  for  incommensurable  powers ;  and  the  whole  theory  of 
logarithms,  so  important,  and  their  use,  so  common,  are  thus  left 
to  rest  on  faith.     In  a  few  cases  new  words  and  new  symbols 

183981 


IT  PREFACE. 

have  been  introduced  :  notably  the  signs  ■•",  ~,  ±,  and  the  copulas 
> ,  <,  3:.    It  is  believed  that  the  need  will  justify  the  innovation. 

Moreover,  the  tendency  of  modern  work  is  to  change  the  tra- 
ditional boundaries  of  algebra  so  as  to  utilize  graphic  represen- 
tation, the  elements  of  infinitesimal  analysis,  and  the  calculus 
of  operations,  and  onl}-  thus  can  the  subject  be  presented  most 
naturall}'  and  philosophically.  A  good  example  is  that  of  the 
so-called  "  imaginaries,"  which,  rightly  presented,  are  as  real 
as -any  other  numbers. 

The  authors  set  out  to  write  a  text-book  for  the  use  of  their 
own  classes  in  the  University,  i.e.,  for  young  men  who  had 
already  studied  the  elements  of  algebra  and  geometry,  and  who 
had  had  some  scholarly  training ;  and,  though  an  elementary  book, 
at  no  time  have  they  thought  of  it  as  a  book  for  beginners.  The 
wants  of  their  classes  have  ever  been  before  them  ;  but  the  work 
has  grown  upon  their  hands  until  it  embraces  many  topics  that, 
from  their  nature  or  their  treatment,  are  quite  beyond  the  range 
of  ordinary  college  instruction.  As  a  text-book,  therefore,  for 
use  in  ordinary  classes  it  must  be  abridged  ;  yet  its  wide  range 
makes  it  all  the  more  valuable  to  teachers,  as  a  book  of  refer- 
ence, and  to  those  bright  scholars  who  wish  either  to  place  their 
knowledge  of  algebra  on  a  sure  foundation,  or  to  make  that 
knowledge  a  stepping-stone  to  the  higher  analysis. 

Many  thanks  are  due  to  Mr.  James' McMahon  and  Mr.  A.  S. 
Hathaway,  instructors  in  mathematics  in  the  University,  for  their 
very  valuable  assistance  in  the  preparation  of  the  text,  and  to 
Mr.  Albert  Jonas  and  Mr.  E.  C.  Murphy,  for  useful  suggestions, 
and  for  help  in  verification  of  the  text  and  in  proof-reading. 

Another  edition  will  contain  chapters  on  :  theory  of  equations, 
integer  analysis,  symbolic  methods,  determinants  and  groups, 
probabilities,  and  insurance  ;  with  a  full  alphabetical  index. 


ooisrTEisrTS, 


I.  — PRIMARY  DEFINITIONS  AND  SIGNS. 

SECTION  PAGE 

1.  Number 2 

2.  Representation  of  Numbers 2 

3.  Positive  and  Negative  Numbers 3 

4.  Special  Signs 6 

5.  Copulas  and  Statements 7 

6.  Addition 9 

7.  Subtraction 10 

8.  Multiplication 11 

9.  Division 13 

10.  Involution 14 

11.  Evolution 15 

12.  Expressions 16 

13.  Functions 19 

14.  Coefficients.  —  Like  and  Unlike  Terms 21 

15.  Degree 21 

16.  lixamples 23 

II.  —  PRIMARY  OPERATIONS. 

1.  Logical  Terms .26 

2.  Combinatory  Properties  of  Operations 27 

3.  Axioms 32 

4.  Addition  Commutative  and  Associative 35 

5.  Sign  of  Product 36 

6.  Multiplication  Commutative  and  Associative 37 

7.  ^lultiplication  Distributive  as  to  Addition 45 

8.  Proportion 48 

9.  Process  of  Addition ^   .     .  52 

10.  Process  of  Subtraction 52 

11.  Process  of  Multiplication 53 

12.  Process  of  Division 60 

13.  Operations  on  Fractions 65 

14.  Examples 67 


VI  CONTENTS. 

ni.  — MEASURES,  MULTIPLES,  AND  FACTORS. 

SECTION  PAGB 

1.  Definitions 79 

2.  Axioms 81 

3.  Measures  and  Multiples 81 

4.  Prime  and  Composite  Numbers.  —  Factors    .    .     .    *.  J^vA*^  83 

5.  Process  of  Finding  the  Highest  Common  Measure  f^...  90 

6.  Process  of  Finding  the  Lowest  Common  Multiple  ....  95 

7.  Process  of  Factoring 96 

8.  Examples 102 

IV. —  PERMUTATIONS  AND  COMBINATIONS. 

1.  Definitions 106 

2.  Permutations 106 

3.  Combinations 112 

4.  Examples 118 

v.  — POWERS  AND  ROOTS   OF  POLYNOMIALS. 

1.  Product  of  Binomial  Factors 121 

2.  The  Binomial  Theorem 122 

3.  The  Polynomial  Theorem 125 

4.  Roots  of  Polynomials 127 

5.  Absolute  and  Relative  Error 129 

6.  Roots  of  Numerals  .     .    .     .    .    .    . 132 

7.  Examples 134 

VI.  — CONTINUED  FRACTIONS. 

1.  Form  of  Continued  Fractions.  —  Convergents 137 

2.  Conversion  of  Common  Fractions 138 

3.  Conversion  of  Surds 141 

4.  Computation  of  Convergents 144 

5.  General  Properties 147 

6.  Secondary  Convergents 153 

7.  Examples 155 

Vn.  — mCOMMENSURABLES,  LIMITS,  INFINITESIMALS, 
AND  DERIVATIVES. 

1.  Variables  and  Constants.  —  Continuity 157 

2.  Incommensurables 159 


CONTENTS.  VU 

SECTION  PAGIS 

3.  Limits 161 

4.  Infinitesimals  and  Infinites 163 

5.  Derivatives 165 

6.  First  Principles 167 

7.  Primary  Operations  on  Incommensurables 168 

8.  General  Properties  of  Limits 169 

9.  General  Properties  of  Derivatives 173 

10.  Indeterminate  Forms 178 

11.  Graphic  Representation  of  Functions 181 

12.  Integration 184 

13.  Examples 189 

VIIL  — POWERS  AND  ROOTS. 

1.  Fractional  Powers 194 

2.  Combinations  of  Commensurable  Powers 195 

3.  Continuity  of  Commensurable  Powers 199 

4.  Incommensurable  Powers 205 

5.  Combinations  of  Powers  in  General 207 

6.  Continuity  of  Powers  in  General 209 

7.  Derivatives  of  Powers 213 

8.  Radicals 215 

9.  Operations  on  Radicals 223 

10.  Examples 227 

IX.  — LOGARITHMS. 

1.  General  Properties 233 

2.  Special  Properties,  Base  10 236 

3.  Computation  of  Logarithms 237 

4.  Tables  of  Logarithms 239 

5.  Operations  with  Common  Logarithms 240 

6.  Examples 250 

X.  — IMAGINARIES. 

1.  Definitions  and  Graphic  Representation 251 

2.  Addition  and  Subtraction 259 

3.  Multiplication  and  Division 264 

4.  Powers  and  Roots 268 

5.  Abridged  Representation 278 

6.  Examples   .     .     .     .' 279 


Vlll  CONTENTS. 

XL  — EQUATIONS. 

SECTION  PAGE 

1.  Statements 281 

2.  Solution  of  Equations.  —  Unknowns 282 

3.  Degree  of  Equation 283 

4.  General  Properties       284 

5.  Simple  Equations  involving  One  Unknown 291 

6.  Elimination 293 

7.  Simple  Equations,  Two  or  More  Unknowns 298 

8.  Graphic  Representation  of  Simple  Equations  involving  Two 

Unknowns 307 

9.  Bezout's ^Method,  Unknown  Multipliers 308 

10.  Special  Problems  of  the  First  Degree 310 

11.  Quadratic  Equations  involving  One  Unknown    ......  313 

12.  Graphic  Representation  of  Quadratic  Equations     ....  318 

13.  Solution  of   Quadratic   Equations  by   Aid  of  Continued 

Fractions 323 

14.  Maxima  and  Minima *" .     .  324 

15.  Simultaneous  Equations 330 

16.  Special  Problems  involving  Quadratics 340 

17.  Binomial  Equations 341 

18.  Logarithmic  and  Exponential  Equations 342 

19.  Examples 343 

XIL  — SERIES. 

1.  Arithmetic  Progression    i 361 

2.  Geometric  Progression 364 

3.  Harmonic  Progression      . 367 

4.  Convergence  and  Divergence 369 

5.  Indeterminate  Series 375 

6.  Imaginary  Series     . - 378 

7.  Expansion  of  Functions  in  Infinite  Series 381 

8.  Method  of  Unknown  Coefl&cients 383 

9.  Binomial  Theorem 390 

10.  Finite  Differences 393 

11.  Interpolation 396 

12.  Taylor's  Theorem 400 

13.  Computation  of  Logarithms 403 

14.  Examples 405 


A  L  G-  E  B  K  A 


I.    PRIMARY  DEFINITIONS   AND   SIGNS. 

Algebra  is  that  branch  of  Mathematics  which  treats  of  the 
relations  of  numbers.  It  is  distinguished  from  Arithmetic,  as 
having  wider  generalizations,  as  using  signs  and  letters  more 
freely,  and  as  recognizing  negatives  and  iraaginaries.  The  ap- 
plications of  many  words  common  in  Arithmetic  are  greatly 
extended  in  Algebra,  and  their  definitions  are  correspondingly 
enlarged. 

The  S3^mbols  explained  below  constitute  a  symbolic  language, 
a  species  of  short-hand  writing,  wherein  numbers  and  their 
relations  are  more  conveniently  expressed  than  in  the  ordinary 
language  of  words.  In  this  language  the  signs  stand  for  words 
and  phrases,  and  generally  haw  11  le  same  grammatical  relations 
as  the  words  and  phrases  themselves.  The  words  may  be 
restored  at  any  time.  The  reader  should  constantl}^  practice 
translating  from  one  form  to  the  other  till  both  are  familiar. 

This  symbolic  language  is  one  of  the  characteristic  features 
of  Algebra ;  and  among  its  man}'  advantages  are  these :  clear- 
ness, brevity,  and  generality  of  statement ;  the  ability  to  mass 
directl}'  under  the  eye,  and  thus  to  bring  before  the  mind  as  a 
whole,  all  the  steps  in  a  long  and  intricate  investigation ;  and 
the  facility  of  tracing  a  number  through  all  the  changes  it  may 
undergo.  Some  other  sciences,  for  example  Chemistry  and 
Logic,  have  a  symbolic  language  of  their  own. 


2  PEIMARY  DEFINITIONS  AND   SIGNS.  [I- 

§  1.     NUMBER. 

'iKineasuping'anj'tbing,  some  unit  of  the  same  kind  is  first 
assu«»6dv  tiiid  the  delation  the  thing  measured  bears  to  this  unit, 
i.e-.,  the  operation' that  if  performed  upon  the  unit  will  produce 
the  given  thing,  is  expressed  by  a  number.  The  unit,  being 
acted  upon,  is  the  operand^  the  number  is  the  operator^  and  the 
thing  produced  is  the  result. 

Such  numbers  are  also  called  abstract  numbers^  because  all 
their  properties  and  relations  are  independent  of  the  particular 
units  used  ;  and  the  units  and  the  measured  things  are  concrete 
numbers.  Abstract  and  concrete  numbers  are  also  called  quan- 
tities. 

Abstract  numbers  likewise  arise  from  the  combination  of 
other  abstract  numbers :  and  in  this  way,  their  relations  form 
the  chief  subject-matter  of  Algebra. 

Two  abstract  numbers  are  equal  if,  operating  upon  the  same 
unit  in  the  same  way,  they  produce  the  same  result. 

An  abstract  number  is  an  integer  if  the  thing  measured  be 
made  up  of  entire  units  ;  a  simple  fraction,  if  the  thing  be  one 
or  more  of  the  equal  parts  that  the  unit  may  be  divided  into. 
Integers,  simple  fractions,  and  such  other  numbers  as  can  be 
reduced  to  integers  or  simple  fractions,  are  commensurable  num- 
bers: those  which  cannot  be  so  reduced  are  either  incommensu- 
rables  or  imaginaries. 

§  2.     REPRESENTATION   OF   NUMBERS. 

Numbers  are  represented  b}'  Arabic  numerals,  or  by  letters. 
Among  the  more  common  forms  are  these : 

0,  1,  2,  3,  . . . ,  10,    read :  naught,  one,  two,  three,  .  ..,ten; 
a,  (3,  y,  8,  €,  0,  TT,  <^,  A,  1,,     read :  alpha,  beta,  gamma,  delta, 
epsilon,  theta,  pi,  phi,  large  delta,  large  sigma; 

A-       X       TT 

-,  -,  -,     read:  four  sevenths,  x  over  y,  half  pi. 

^   y  ^ 

a',  b",  &^,  f^"*^,     read  :  a  prime,  b  second,  c  fourth,  d  m'*; 
Poi  P\^  2^x1     read :  p  sub  zero,  p  sub  one,  p  sub  x. 

0  ctO 


§  3.]  POSITIVE   AXD   NEGATIVE   NUMBERS.  3 

The  accents,  numerals,  and  letters,  attached  to  other  numerals 
and  letters,  are  indices.     An  index  attached  below  its  letter  is  a 
suffix  or  subscript^  and  is  read  siib.    The  index  of  a  power  [§10] 
is  an  exponent.     The  letter  or  numeral  to  which  the  index  is 
attached  is  the  stem.    Sometimes  the  indices  are  written  without 
the  stem  ;  this  form  of  writing  is  called  the  umhral  notation. 
E.g.,  instead  of  ai,  2 ;  «3, 4 ;  •  •  •  «i,  s,    write  1,2;  3,  4  ;.../,  7i. 
The  accent  and  subscript  notation  has  two  chief  advantages  : 
It  gives  a  very  great  number  of  distinct  symbols. 
It  permits  numbers  analogously  related  to  the  problem 
in  hand  to  be  represented  by  the  same  letters. 
Eg-y  P\  P"^  P'"^     or     Pi,  2^2,  ih,  may  stand  for  the  princi- 
pals of  three  promissor}-  notes  ; 
then       t\  t'\  t'",      or     ^„  t^,  ^3,  will  naturally  stand  for  the 
three  times  for  which  these  three  notes  are  respec- 
tively given, 
and        r\  r",  r'",     or     ?'i,  rs,  rg,  for  the  three  rates. 

The  value  of  a  letter  or  other  S3'mbol  is  the  number  for  which 
it  stands.  Ordinarily  the  same  letter  stands  for  but  one  number 
during  an}'  one  investigation,  but  for  different  numbers  in  differ- 
ent investigations  ;  and  different  letters,  or  the  same  letter  with 
different  indices,  for  different  numbers  in  the  same  investigation. 

§  3.     POSITIVE    AND    NEGATIVE   NUMBERS. 

When  the  measuring  unit  is  taken  in  the  same  sense  as  the 
quantity  measured,  the  number  is  positive;  when  in  the  opposite 
sense,  the  number  is  negative.  In  which  sense  the  unit  shall  be 
taken,  is  a  matter  of  custom  or  convenience. 

Manifestly,  if  two  quantities,  opposite  in  sense,  are  measured 
by  the  same  unit,  one  number  is  positive  and  the  other  negative  : 

E.g.,  if  distances  to  the  north  or  east  from  a  given  point  are 
positive,  distances  to  the  south  or  west  are  negative :  2. e.,  if 
the  measuring  unit  is  a  northerly  or  easterly  unit,  then  southerly 
or  westerly  distances  are  expressed  by  negative  numbers,  and 
vice  versa. 


4  PRIMARY  DEFINITIONS   AND   SIGNS.  [I. 

So,     if  the  revolutions  of  a  wheel  forward  are  positive,  revo- 
lutions backward  are  negative ; 
if  assets  are  positive,  liabilities  are  negative  ; 
if  dates  a.d.  are  positive,  dates  B.C.  are  negative; 
if  the  readings  of  a  thermometer  above  zero  are  positive, 
the  readings  below  zero  are  negative. 

Tlie  primary  notion  of  a  negative  number  is  that  of  one  which, 
when  taken  with  a  ix>sitive  number  of  the  same  kind,  goes  to 
diminish  it,  to  cancel  it  altogether,  or  to  reverse  it. 

E.g.^  liabilities  neutralize  (negative)  so  much  of  assets, 
thereb}'  diminishing  the  net  assets  or  leaving  a  net  liabilit}'. 

If  two  numbers  of  the  same  kind,  whin  taken  together,  exactly 
cancel  each  other,  the}'  are  opposites^  one  of  the  other. 

Manifestl}',  of  two  opposites,  one  is  positive  and  the  other  is 
negative. 

So,  if  numbers  are  used  as  indices  of  two  algebraic  operations 
which  when  performed  successive!}'  tend  to  neutralize  each  other, 
a  positive  number  is  commonl}'  used  for  one  index  and  a  nega- 
tive number  for  the  other ;  and  sometimes,  as  with  exponents 
of  powers  [§  10],  custom  has  permanently  determined  which 
index  shall  be  ix)sitive  and  which  negative. 

When  denoted  b}'  Arabic  numerals,  positive  numbers  are  writ- 
ten with  the  sign  +  or  with  no  sign,  and  negative  numbers  with 
the  sign  — ,  and  it  is  evident  at  sight  whether  the  number  is 
positive  or  negative. 

E.g.^  if  the  measuring  unit  be  $1  of  assets,  then  +  100,  or 
simply  100  without  the  sign,  expresses  the  net  value  of  an  estate 
whose  assets  exceed  its  liabilities  b}-  $100 ;  and  —100,  that  of 
an  estate  whose  liabilities  exceed  its  assets  by  $100. 

But,  if  a  number  be  denoted  by  a  letter,  it  is  not  evident  upon 
its  face,  and  often  it  is  not  necessar}'  to  know,  whether  that  let- 
ter denotes  a  positive  or  a  negative  number. 

E.g.^  in  the  above  example,  n  may  stand  either  for  -f  100  or 
for  —  100,  at  the  pleasure  of  the  writer.  If,  however,  n  stands 
for  4-100,  then  —  n  stands  for  — 100  ;  and  if  n  stands  for  —100, 
then  —  N  stands  for  -f  100.  In  either  case  -f-  n  and  —  n  ai'c 
opposites. 


§  3.]  POSITIVE   AXD    NEGATIVE   NUMBERS.  5 

In  this  use  of  the  signs  -j-  and  — ,  the}'  are  called  signs  of 
quality,  since  the}'  indicate  the  qualit}',  in  an  important  particu- 
lar, of  the  quantities  measured,  and  of  the  numbers  before  which 
they  stand. 

These  signs  are  also  used  to  indicate  the  operations  of  addition 
and  subtraction  [§§  6,  7],  and  are  then  called  sigyis  of  operation ; 
but,  as  the  reader  will  see  wlien  he  comes  to  the  study  of  these 
operations,  the  two  uses  are  always  in  accord,  and  the  signs  may 
often  be  understood  in  either  wa}-  at  pleasure.  Sometimes  signs 
performing  both  offices  occur  before  the  same  number  [§§  6,  7]. 

The  sign  ■•"  before  a  number  denotes  either  the  number  itself, 
or  its  opposite,  whichever  of  them  is  positive  ;  the  sign  "  denotes 
whichever  of  them  is  negative  ;  i.e.,  a  number  preceded  b}'  "•"  is 
essentially  positive,  and  a  number  i)receded  b}'  ~  is  essentially 
negative. 

E.g.,  if  N  stands  either  for  100  or  for  —100,  +x,  read  n  taken 
positive,  stands  for  -f-100  ;  and  ~n,  read  n  taken  negative,  stands 
for  -  100. 

So,  +100  may  always  be  written  for  + 100,  and  "100  for  — 100 ; 
but  not  ■'"N  for  n  or  +n,  nor  "n  for  —  n,  unless  the  value  of  n 
be  positive. 

Manifestly,  +  N  and  ~n  are  opposites ;  and  so  are  +100  and  "100. 

N'oTE.  —  The  reader  should  observe  that  some  things  admit 
of  negatives  and  some  do  not. 

E.g.,  time  may  be  counted  backwards  as  well  as  forwards 
from  any  given  date ;  so  may  distance  from  any  given  point ; 
so  may  heat  and  cold  from  an  arbitrary  zero ;  so  may  money  of 
account,  as  above ;  but  with  real  dollars,  say  five  of  them,  he 
will  find,  when  he  tries  to  count  past  none,  —  five,  four,  three, 
two,  one,  none,  —  that  he  is  attempting  to  do  what  is  impossible. 

So,  when  he  comes  to  the  study  of  the  so-called  imaginaries, 
he  will  find  that  for  some  things  they  have  a  real  existence,  but 
for  other  things  they  have  not. 

So,  for  some  things,  fractions  have  no  existence. 

E.g.,  ^  ot  a  man,  or  J  of  an  atom,  or  1^  events  or  facts, 
would  be  unmeaning. 


6  PKIMARY  DEFINITIONS   AND   SIGNS.  [I. 

§  4.     SPECIAL    SIGNS. 

The  sign  of  continuation  is  . . . ,  read  and  so  on. 

E.g.,  1,  -2,  +3,   ...,  +9 
means     1,  —2,  +3,  —4,  +5,  — G,  +7,  —8,  +9. 

The  signs  of  inference  are  •.-,   read  since  or  because^ 
and  .-.,  read  therefore. 

E.g.,  '.'  80    cts.<$l,       .-.  400cts.<$5; 
or  400ets.<S5,       •.•     80cte.<$l. 

The  signs  of  grouping  are  (),[],  j  J,  ,  |  .  The}'  show 
that  all  within  the  brackets^  under  the  horizontal  bar^  or  before 
the  vertical  bar,  is  taken  together  as  one  number,  and  subject  to 
the  same  operation;  viz.,  that  which  is  indicated  b}'  the  sign 
preceding  or  following  it,  or  by  the  index  attached  to  it. 

When  two  or  more  numbers  joined  b}-  the  signs  -f-  and  — 
[§§  6,  7]  are  grouped  together  b}'  a  bar  or  brackets,  the}-  form 
an  aggregate. 

E.g.,  (l+2-i-3)x5  — 2  is  the  product  of  two  aggregates  [§  8]. 

When  two  statements  are  identical,  except  only  for  a  few 
characteristic  words  or  signs,  then,  as  a  matter  of  convenience, 
the  two  statements  ma}'  be  written  together  as  a  double  state- 
ment, b}'  placing  the  pairs  of  corresponding  words  or  signs  one 
above  the  other. 

E.g.,  •••  the  battle  of  Salamis  was  fought  480  b.c.  and  that  of 

AYaterloo  1815  a.d., 

^    Salamis    _„„  ^^„„V4.  ooorc   ,.„    i  before  Waterloo. 
•  •  Waterloo  ^''^  ^""8^'  ^^95  yrs.  {  ^^^,  ^^^^^^.^ 

So,      it^  ^"^  30° ^  °''^'^<='"  .Vf  tei-day  ^^      ,  to-day 

'is  '  warmer  to-da}'  '  yesterday. 

In  such  double  statements,  all  the  words  and  signs  in  the 
upper  line,  together  with  the  common  parts,  go  to  make  up  the 
first  statement ;  and  all  the  words  and  signs  in  the  lower  line, 
together  with  the  common  parts,  the  second  statement.  In  the 
same  waj',  three  or  more  statements  may  be  written  together. 

^\"lien,  of  a  double  statement,  onl}-  one  part  can  be  true,  but 
which  that  is,  is  unknown,  such  statement  is  ambiguous. 


§5.]  COPULAS   AND   STATEMENTS.  7 

§  5.     COPULAS  AND   STATEMENTS. 

Two  numbers  are  equal  when, 'in  every  combination  which 
contains  either  of  them,  the  other  may  take  its  place  without 
chanojinor  the  result. 

When  one  number  is  equal  to  another,  the  two  are  joined  by 
the  sign  =,  read  equals^  or  is  equal  to,  and  the  whole  is  an 
equation;  or  b}'  the  sign  =,  read  is  identical  ivith,  and  the  whole 
is  an  identity. 

E.g.,  100  cents  =  1  dollar  ;  100  cents  =  100  cents  ;  x  =  a;. 
An  identit}'  is  an  equation  wherein  the  two  numbers  remain 
equal,  however  the  values  of  an}'  of  the  letters  may  change. 
Ever}'  identity  is  an  equation,  but  not  ever}-  equation  is  an 
identit}'.  Hence  =  ma}'  always  take  the  place  of  = ,  but  =  not 
always  of  = . 

The  sign  =  is  also  used  for  "  stands  for"  and  "represents." 
A        E.fj.,   p  =  principal,  t  =  time,  r  =  rate,  i  =  interest. 
/         When  one  number  is  not  equal  to  another,  the}'  are  joined  by 
V       the  signs  ^.^  ^,  <,  >,  <,  >,  read :  not  equal  to,  not  identical 
y7  tvith,  less  than,  greater  than,  smaller  than,  larger  than. 
^  E.g.,  80cts.^$l,  lOOcts.^^l,  80cts.<$l,  120cts.>$l, 

C^  80cts.<$l,  120cts.  >S1. 

:>^r        So,      <,  >,  1^,  ^,  mean  not  less  than,  not  greater  than,  etc. 

The  words  "greater"  and  "  less"  are  here  used  in  a  technical 
sense,  and  may  be  expressed  by  higher  and  loiaer  in  speaking  of 
,,  temperatures  and  elevations,  by  north  of  or  east  of  and  south  of 
or  ivest  of  in  Surveying  and  Geography,  by  later  and  earlier  in 
comparing  two  dates,  and  so  on ;  but  "  larger"  and  "  smaller" 
take  account  of  the  size  of  the  two  numbers  only. 

E.g.,  30  ft.  up  >  50  ft.  down,  and  30  ft.  down  >  50  ft.  down  ; 
i.e.,      +30>-50,     and -30  > -50. 

But  +  30  <  -50,     and  "30  <  "50. 

If  two  numbers  be  equally  large,  the  sign  is  :e:  ;  if  not  equally 
large,  ^. 

^.^., +1600:^-1600;  +1600:^-1700. 


8  PRrMARY  DEFINITIONS   AND   SIGNS.  [I. 

In  general,  an}'  positive  number,  however  small,  is  greater 
than  any  negative  number,  however  large  ;  and,  of  two  negative 
numbers,  the  smaller  is  greater  than  the  larger. 

Manifestly,  the  greater  a  number  the  less  is  its  opposite  ;  but 
a  number  and  its  opposite  are  equally  large. 

The  signs  =,  =,  -,  4-,  ^,  ^,  <,  >,  <,  >,  <,  >,  ^,  ^ 
are  signs  of  assertion  or  copulas. 

Equations,  identities,  and  inequalities  are  statements^  and  when 
of  general  truths,  the}'  are  formulae.  The  first  member  is  all  that 
precedes  the  copula,  and  the  second  member^  all  that  follows  it. 

A  continued  statement  is  one  having  more  than  two  members ; 
it  is  equivalent  to  as  man}'  simple  statements  as  there  are  copu- 
las, and  each  copula,  unless  preceded  by  a  comma,  connects  the 
two  members  immediately  adjacent  to  it. 

E.g.,     l<3<5<-7^9 
is  equivalent  to  the  group  of  independent  statements 
1<3,  3<5,  5<-7,  -7^9. 

So,-.*a<6         .-.  2a<26<36 
is  equivalent  to  the  chain  of  connected  statements 

•.•  a<6         .-.  2a<26,  and  26<36,  and  .-.  2a<36. 

But-.*  a<&         .*.  o,  <2a,  <26 
is  equivalent  to  the  chain  of  connected  statements 

•.•a<6         .-.  a  (which  <2a)  <  2  6, 
and  is  read:  Since  a  is  smaller  than  6,  therefore  a,  which  is 
smaller  than  2  a,  is  smaller  than  26. 

This  is,  in  effect,  a  brief  form  for  a  logical  chain  of  statements. 
The  office  of  the  commas  is  to  parenthesize  what  is  between 
them,  and  compare  directly  what  precedes  the  first  comma  and 
what  follows  the  last  comma ;  the  basis  of  comparison  being 
found  in  what  the  commas  enclose.  The  first  comma  is  read  which. 

So,       a^  —  a^  =6, 
is  equivalent  to  the  two  independent  statements 

a^  —  a     and     a  =  b; 
and  here,  too,  the  office  of  the  comma  is  to  carry  forward  the 
first  member,  a,  and  compare  it  with  b  which  follows  the  comma. 
The  comma  is  read  a7id. 


§  6.]  ADDITION.  9 

§  6.     ADDITION. 

The  sum  of  two  or  more  concrete  numbers  of  the  same  kind 
is  a  new  concrete  number  got  by  joining  together  the  several 
things  measured,  and  then  measuring  the  aggregate  by  the  same 
unit  that  measured  the  original  numbers. 

The  sum  of  two  or  more  abstract  numbers  is  a  new  abstract 
number  which,  if  used  as  an  operator  upon  any  unit,  will  give 
the  same  result  as  if  the  original  numbers  were  first  used  as 
operators  upon  the  unit  and  their  results  were  then  added. 

Addition  is  the  process  of  finding  the  sum  of  two  or  more 
numbers.  If  the  numbers  added  be  commensurable,  then,  at 
bottom,  addition  is  but  counting  either  by  entire  units  or  by  the 
aliquot  .parts  of  a  unit :  on  (forward)  if  positive  numbers  be 
added ;  off  (backward)  if  negative  numbers  be  added. 

The  sign  of  addition  is  +  ;  read  phis,  or  the  sum  of ...  and  ... 

E.g.,    oOcts.  +  GOcts. +  90cts.  =$2;    50  +  60  +  90  =  200. 

In  Algebra  the  word  "addition"  is  used  in  a  broader  sense  than 
in  Arithmetic,  and  covers  negative  as  well  as  positive  numbers. 

E.g.,    he  who  has  $10,000  cash  and  $4,000  debts  is  worth 
but  $6,000; 
i.e.,  $10,000  cash  +  $4,000  debts  =  $6,000  net  assets  ; 

+10,000+-4,000  =  +G,000. 

So,        a  train  which  has  run  east  10  miles,  then  west  20  miles 
over  the  same  track,  is  10  miles  west  of  the  start- 
ing-point ; 
i.e.,  10  east-miles  +  20  west-miles  =  10  west-miles  ; 

+10 +-20  =-10. 

But      a  train  which  has  run  west  10  miles,  then  west  20  miles 
more,  is  30  miles  west  of  the  starting-point ; 
i.e.,  10  west-miles  +  20  west-miles  =  30  west-miles  ; 

-10 +-20  =-30. 

Though  the  numbers  to  be  added  must  always  be  of  the  same 
kind,  they  are  often  expressed  by  letters  whose  values  are  not 
known,  or  in  units  whose  values  are  different,  and  which  there-, 
fore  cannot  be  reduced  to  one  sum. 

E.g.,    5^33'°30''  +  12H7'"30«  =  18^21'". 


10  PRIMARY  DEFINITIONS   AND   SIGNS.  [I. 

Manifesth',  the  sum  of  two  opposites  is  0. 
E.g.^    90  ft.  up  is  the  opposite  of  90  ft.  down  ; 
i.e. ,      +  90  is  the  opposite  of  —  90  ;  and  the  sum  of  the  two  is  0. 
So,   -fa  and  —a,     "ct  and  +a,     26  — 3  c  and  3  c— 26. 


§7.    SUBTRACTION. 

Subtraction  is  the  inverse  of  addition,  and  consists  in  find- 
ing what  number  must  be  added  to  one  number,  the  subtra- 
hend^ to  give  another  number,  the  minuend.  The  result  is  the 
remainder.,  and  the  sign  is  — ,  read  minus  or  the  excess  of  ... 
over  ...  One  or  both  of  the  numbers  may  be  negative,  and  the 
minuend  may  be  less  than  the  subtrahend. 

E.g.,    S50-$40=  610,       S40-   S50=-$10. 
-$50  --840  ="6 10,     -$40  -  -$50  =+$  10. 

So,       if  of  two  men  A  has  $10,000  cash  and  no  debts,  and 
B  has  $5,000  debts  but  no  assets, 
then         A  is  $  15,000  better  off  than  B,  , 

i.e.,        +10,000  --5,000  =-»-15,000  ; 
and  B  is  $  15,000  worse  off  than  A, 

i.e.,        -5,000  -+10,000  =-15,000. 

So,  *.*  the  battle  of  Salamis  was  fought  480  b.c,  and  that  of 
Waterloo  1815  a.d., 
.*.  Waterloo  was  fought  2295  ^ears  after  Salamis, 
i.e.,        +1815  --480  =+2295  ; 

and  Salamis  was  fought  2295  years  before  Waterloo, 

i.e.,        -480  -+1815  =-2295. 

So,       if  to-da}^  a  thermometer  read  10°  below  zero,  and  yes- 
terday it  read  20°  below  zero, 
then         it  is  10°  warmer  to-day  than  yesterday, 
i.e.,        -10 --20  =+10; 

and  it  was  10°  colder  yesterday  than  to-day, 

i.e.,        -20 --10  =-10. 

The  difference  between  two  numbers  is  the  remainder  found 
by  subtracting  the  less  from  the  greater ;  the  sign  is  ~ . 

E.g.,    16  — 12  =  12-- 16  =  4;    "16 '-+12  =+16 ---12  =  28. 


§  8.1  MULTIPLICATION.  11 


§  8.     MULTIPLICATION. 

The  product  of  a  concrete  number,  the  multiplicand ^  by  an 
abstract  number,  the  multiplier,  is  a  concrete  number  of  the 
same  kind  as  the  multiplicand,  and  bearing  to  the  multiplicand 
the  same  relation  as  the  multiplier  bears  to  unity. 

Tlie  product  of  two  or  more  abstract  numbers  is  a  new 
abstract  number  such  that,  if  a  unit  be  multiplied  by  it,  the 
product  is  the  same  as  the  final  product  obtained  by  multiplying 
the  unit  by  the  first  of  the  numbers,  the  product  so  found  by 
the  second  of  thfem,  and  so  on. 

Multiplication  is  the  process  of  finding  the  product  of  two  or 
more  numbers  ;  the  numbers  are  the  factors  of  the  product. 

Multiplication   by   a    -{  ^        .       integer  is   but   a   repeated 

.  addition       ^^  ^^^^  multiplicand  ^  *^        0,  and  multiplication 
'  subtraction  '  from 

by  a   J  P°^'«7''    fraction   is   fte  repeated  J  ^'1'^"'°°  '" 

'  negative  '  subtraction  from 

0  of  the  equal  parts  into  which  the  multiplicand  is  divided. 

In  the  last  analysis,  multiplication  is  but  a  counting,  on  or 

off,  according  as  the  multiplier  is  positive  or  negative  ;  but  it  is 

a  counting  by  groups,  each  equal  to  the  multiplicand,  if  the 

multiplier  be  an  integer,  and  by  aliquot  parts  of  such  groups  if 

it  be  a  fraction,  instead  of  by  single  units  as  in  addition. 

E.g.,    five,  ten,  fifteen,  twenty,  twenty -five,  thirty,  gives  the 

product  of  five  by  six,  or  of  "five  by  "six. 

So,       "five,  "ten,  "fifteen,  gives  the  product  of  "five  by  three, 

or  of  five  by  "three. 

So,       one  half  of  five,  two  halves  of  five,  three  halves  of  five, 

gives  the  product  of  five  by  f ,  or  of  "five  by  "f . 

So,       i  of  f ,  I  of  f ,  -3-  of  \,  gives  the  product  of  -f-  b}'  f ,  or 

of-fby-|. 

So,       f ,  f ,  f  gives  tlie  product  of  |  by  3  or  of  "f  by  "3. 


12  PRIMARY   DEFINITIONS   AND   SIGNS.  [I. 

The  signs  of  miiltiplicatiou  are  x ,  read  hy^  and  • ,  read  into. 

E.g.,    50  cts.  X  8  =  $4;  8-50  cts.  =  84. 

So,  placing  the  factors  one  after  the  other,  with  no  sign  be- 
tween them,  means  multiplication  of  the  first  into  the  second,  or 
of  the  second  b}'  the  first. 

E.g.,    ah  is  the  product  of  a  into  6,  or  of  h  hy  a, 
and  ab  =  a'b  =  b  X  a. 

When  the  product  of  two  numbers  is  multiplied  b^'  a  third 
number,  such  multiplication  is  the  continued  multiplication  of  the 
tlu'ee  numbers ;  so  for  four  numbers,  for  five  numbers,  and  so 
on ;  and  the  product  of  such  multiplication  is  the  continued 
product  of  the  several  factors. 

E.g.,    5xGx  7  =  210,     and     5- 6- 7  =  210. 

The  continued  product  of  the  natural  numbers  1  •  2  •  3  ...  is 
indicated  by  the  sign  !  placed  after  the  last  factor,  or  by  the 
sign  |_  placed  before  and  under  the  last  factor. 

E.g. ,    5  !  or  [5^,  read  factorial  5,  =l-2-3'4'5,  =120; 
n  !  or  [n,  read  factorial  n,  =  1 '  2  •  S  - . . .  n. 

Some  peculiar  properties  of  negatives  appear  in  multiplication. 

E.g.,    if  a  train,  now  at  a,  is  running  east  20  miles  an  hour, 
then         five  hours  hence  it  will  be  100  miles  east  of  a, 
i.e.,        +20  x+5=+100; 

but  five  hours  ago  it  was  100  miles  west  of  a, 

i.e.,        +20  x -5  =-100. 

So,       if  the  train  is  backing,  i.e.  running  west, 
then  five  hours  hence  it  will  be  100  miles  west  of  a, 

i.e.,        -20  X +5  =-100; 

but  five  hours  ago  it  was  100  miles  east  of  a, 

i.e.,        -20  x-5=+100. 

Two  numbers  whose  product  is  1  are  reciprocals  of  each  other. 

E.g.,    4  is  the  reciprocal  of  J  ;   —  3  of  —  i ;   J  of  f . 

Manifestly,  the  larger  a  number,  the  smaller  its  reciprocal. 

The  product  of  a  number  by  an  integer  is  a  multiple  of  that 
number:  the  double,  triple,  quadruple,  •••,  when  the  multiplier 
is  2,  3,  4,  .... 


§  9,]  DIVISION.  13 

§  9.     DIVISION. 

Division  is  the  inverse  of  multiplication,  and  consists  in  find- 
ing either  factor,  when  the  product  and  the  other  factor  are 
given.  The  product  is  now  called  the  dividend,  the  given  factor 
is  the  divisor^  and  the  result  is  the  quotient. 

E.g.,  '.'  the  product  of  5  b}*  10  is  50, 

.-.  the  quotient  of  50  hy  {  jj^^* 
So,      •••  the  product  of  a  into  h  is  ah, 
.'.  the  quotient  of  ah  by  ^  J^  [^  ^• 

"^^^^^  dhlsoi"  '  ^^"^o  *^^^  mnltii^lier,  is  an  abstract  number 
[§  8]  ;  and  the  {  ^^^^^^^^^^  ^^^^  (lividena  arc  alike  in  kind.    When 

both  factors  are  abstract,  the  two  definitions  of  division  agree, 
as  will  appear  later. 

E.g.,  •.'  the  product  of  $5  b}'  4  is  $20, 

.-.  the  quotient  of  S20  by^  \\^\\' 

The  signs  of  division  are  :  ,  read  the  ratio  of ...  to  ...,  and  -t-, 
read  divided  by,  or  the  quotient  of  ...by 

E.g.,   $20:85  =  4;    S20-f-4  =  $5;    20:5  =  4;    20-^4  =  5. 

So,  writing  the  dividend  over  the  divisor  with  a  horizontal  line 
between  them  means  division.  The  dividend  is  then  called  the 
numerator,  the  divisor  the  denominator,  and  the  whole  expression 
a  fraction.  Hence  a  fraction  is  the  expression  for  the  quotient 
in  a  division  as  3'et  unperformed. 

Note.  This  definition  of  a  fraction  differs  from  that  hereto- 
fore given  [§  1],  but  later  it  will  appear  that  the  two  definitions 
are  in  full  accord. 

If  the  dividend  be  a  multiple  of  the  divisor,  then  the  quotient 
is  an  integer  and  the  division  is  complete ;  but  if  the  dividend 
be  not  a  multiple  of  the  divisor,  its  excess  over  the  greatest 
multiple  that  is  contained  in  it  is  the  remainder. 

E.g.,    27  :  5  =  5,    quotient  with  2  remainder. 


a.        ^ 

14  PKIMARY  DEFINITIONS  AND   SIGNS.  [I- 

§10.     INVOLUTION. 

A  ^  i^^^"^y^    inteqral  power  of   a  number  is  the  continued 
»  negative         -^       ^ 

^  ^"^^ti" n\  ^^  "°^^^  ^^  ^^^  ^^^^"  number. 

Tho  number  whose  power  is  sought  is  the  base. 

The  symbol  that  shows  how  many  times  the  base  is  used  as 
^  multipner  .^  ^^^  exponent;  it  is  written  at  the  right  and  above 

the  Z!  and  is  ^  ^^  for  a  ^  ^^  power. 

E.g.^    lxaxaxa  =  a^,  read 

third  x)ower  of  a,  a  third  power ^  or  a  cwfee. 
1  X  a  X  a  =  a^,  read 

second  power  of  a.^  o.  second  power  ^  or  a  square. 
1  X  a  =  a^  read 

first  power  of  a,  a  first  power,  or  simply  a. 
1  =  a°,  read 

zeroth  power  o/  a,  a  zeroth  power. 
1  -J-  a  =  a~\  read 

mimis  Jirst  2)ower  of  a,  or  a  minus  first  power. 
1  -T-  a  -T-  a  =  a~^,  read 

minus  second  power  of  a,  or  a  minus  second  power. 
1 -f-a-^a-=- a=  a~^,  read 

minus  third  power  of  a,  or  a  minus  third  power. 
A  ?'ooi  of  a  number  is  one  of  the  equal  factors  into  which  it 
may  be  resolved.     The  number  whose  root  is  sought  is  the 
base ;  the  symbol  that  shows  into  how  many  equal  factors  the 
base  is  resolved  is  the  root-index.     The  radical  sign,  -yf,  is  writ- 
ten before  the  base,  and  the  root-index  is  at  the  left  and  above 
it ;  or  else  the  reciprocal  of  the  root-index  is  attached  to  the 
base  as  an  exponent.     The  root-index  2  need  not  be  written. 
E.g.,    -^4,  or  simply  V^  =  ^  =  -2  ;  -^243  =  243*  =  7. 
A  fractional  p>ower  of  a  number  is  either  a  root  of  the  number 
or  some  integral  power  of  such  root.     The  exponent  is  then  a 
simple  fraction  whose  denominator  shows  into  how  many  equal 


§  11]  EVOLUTION.  15 

factors  the  base  is  resolved,  and  whose  numerator  shows  how 

many  times  one  of  these  factors  is  used  as  -^   t  • 
•^  '  divisor. 

E.g.^    64"*  =1x4x4  =  16,  read  64,  I  power,  equals  16. 

64~^=  1  -^  4  -i-  4  =  Jg,  read  64,  —^^ power,  equals  Jg-. 
So,       a"  =  Va»  c^  =  {^c)\  a;?  =  (V^)^  'k~l=:{^'k)-K 

The  words  "  integral,"  "  fractional,"  ''  positive,"  and  "  nega- 
tive "  apply  to  the  exponents  only,  and  not  at  all  to  the  results 
of  the  operations  indicated;  i.e.,  a  positive  integral  power  is 
one  whose  exponent  is  a  positive  integer,  and  so  on. 

Integral  and  fractional  powers  are  commensurable  powers. 
Those  powers  whose  exponents  are  incommensurable  are  called 
incommensurable  powers ;  they  are  defined  in  [VIII.  §  4]. 

Involution  is  the  process  of  finding  the  powers  of  numbers ; 
its  sign  is  the  position  of  the  exponent. 

Note.  — The  reader  may  compare  what  is  here  said  of  posi- 
tive and  negative  exponents,  as  indices  of  repeated  multiplica- 
tion and  division  of  a  unit  by  the  base,  or  by  one  of  the  equal 
factors  of  the  base,  with  what  is  said  in  §  3  of  operations  which 
tend  to  neutralize  each  other.  He  will  then  see  the  peculiar 
propriety  of  expressing  repeated  multiplication  by  a  positive 
exponent,  and  repeated  division,  the  inverse  of  multiplication, 
by  a  negative  exponent. 

§  11.     EVOLUTION. 

Evolution  is  the  inverse  of  involution,  and  consists  in  finding 
a  base  that,  when  raised  to  the  power  denoted  by  the  index, 
produces  the  given  number.     The  result  is  the  root. 

The  logarithm  of  a  number  is  the  exponent  of  that  power  to 
which  a  base  must  be  raised  to  give  the  number.  The  finding 
of  logarithms  is  another  inverse  of  involution. 

E.g.,  '.'  10-  =  100,  .-.  2  is  the  logarithm  of  100  taken  to  the 
base  10  ;  it  is  written  logio  100  =  2,  and  read  log, 
base  10,  o/lOO  equals  2. 

So,       logiol000  =  3,  logiolO  =  l,  logiol  =  0,  logio.l=-l. 


16  PRIMARY  DEFINITIONS   AND   SIGNS.  [I. 

§  12.     EXPRESSIONS. 

An  Algebraic  Expression  is  a  number  or  combination  of 
numbers  written  in  algebraic  form.  It  is  called  an  "expression" 
or  a  "  number,"  according  as  the  thought  is  of  the  S3'mbol  or  of 
the  value  which  the  S3'mbol  represents. 

Unless  a  single  letter  or  numeral,  an  expression  is  made  up  of 
simpler  expressions  affected  or  combined  by  signs  of  operation  ; 
and  the  order  of  these  operations  is  as  follows  : 

1.  Everj^  letter  or  numeral,  with  its  indices,  if  an}',  denotes 
a  number  b}'  itself;  and  so  does  ever}'  expression  united  by  a 
bar  or  parenthesis.  These  numbers,  in  turn,  may  be  affected 
b}'  exponents,  etc.  ;  but  each  exponent  affects  onl}'  the  single 
numeral,  letter,  or  parenthesis  it  is  written  to ;  and  if  a  power 
of  a  power  is  to  be  denoted,  the  new  base  must  be  parenthesized. 

E.g.,    2^^^a^b\x-y){x  +  yy  is  the  product  of  2^  3^,  a^  h*, 
(x-y),  and  {x-hyY; 
but  \_{2^syaY  is  the  cube  of  the  product  of  a  by  the  square 

of  233. 
So,       {a^y  is  the  cth  power  of  a*;  but  a^"  is  the  b'th  power  of  a. 

So,        a**   is  the  b"  th  power  of  a. 

2.  When  a  product  is  denoted  by  writing  the  factors  together 
without  the  sign  X  or  • ,  or  when  a  quotient  is  denoted  by  a 
fraction,  the  product  or  quotient  is  affected,  as  a  single  number, 
by  the  adjacent  sign  V?  ^^Si  >^'>  '•>  ^  ^5  +?  —  ?  "^i  or  ~. 

E.g.,    ^2 ab  -a^y^-.S*  denotes  that  positive  the  square  root  of 
2ab  is  multiplied  into  ay^y^,  and  the  product  di- 
vided by  3* ; 
but  ^ 2  ab  '  our  if :  3*  is  any  square  root  of  2  ab  -x^y^:  3'*. 

So,       logfy^  is  the  logarithm  of  ^y^ ; 
but  log  f  •  ^  is  the  product  of  log  J  into  ?/^. 

3.  In  this  book,  when  successive  numbers  are  separated,  some 
by  the  sign  X ,  •,  :,  or  -^,  and  some  by  +  or  — ,  the  multiplica- 
tions and  divisions  are  first  performed,  and  then  the  products  and 
quotients  are  added  or  subtracted  ;  and  if  several  of  these  signs  of 


§  12.]  EXPEESSIONS.  17 

multiplication  and  division  occur  in  succession,  or  several  signs 
of  addition  and  subtraction,  the  left-hand  operation  is  first  per- 
formed.    But  the  usage  among  algebraists  is  not  uniform. 

E.g.,    3:2.6-6^3x2  +  1  denotes  that  from  f  •  6  is  sub- 
tracted |X2,  and  to  the  remainder,  5,  is  added  1. 
Those  parts  of  an  expression  which  are  joined  by  the  signs  -f- 

or  -  are  terms,  and  terms  are^  '^"'^f  ^'  which  ^  ^^"^^7       ,  . 

'  simple  '  do  not  contam 

the  sign  +  or  —  except  in  an  index. 

An  expression  of  one  term  onl}'  is  a  monomial,  of  two  terms 

a  binomial,  of  three  terms  a  trinomial,  of  four  terms  a  quadri- 

nomial;  of  two  or  more  terms  a  polynomial. 

.  •       •    »  numerical     i       -,  i  , 

An  expression  is<[  7-.      i         when  the  numbers  are  expressed 

I  wholly  by  numerals ;  ,  -finite         ,        .,  i         /. 

-!  wholly  or  in  part  by  letters ;    ^ivfintte  '"''"'"  "''^  ""'"'^''■'  °^ 

operations  implied  is  {    "^!  ^.'  , 
^  ^  '  unl united. 

A  finite  expression  is<|  f,?^^^^^^  when  there  is  implied 

\  other  operation  than  addition,  subtraction,  multiplication, 

division,  and  involution  to  commensurable  powers. 

An  algebraic  expression  is  ^  '  '^  ^^^^  when  it  i  ^^^       be  freed 

'  surd  '  cannot 

*^ ^.       .entire  or  integral    ,        .free  from  divisors  and  roots. 

from  roots  ;<^     ,.       ,     '^       when<      ,  «        -         ,.  • 

'  ^jractional  'not  free  from  divisors. 

E.g.fBbc,  A,  A^'^^iX""^^  [r  being  a  positive  integer]  are  entire 

simple  rational  monomials. 

So,    -^ — ,      *  •    ,  ,    (a  +  a;"-')  are  complex  fractional 

a         a-\-l     a—1 

monomials. 
a  +  a;"^  is  a  fractional  binomial  with  simple  terms. 

(a-\-b)-\ is  a  binomial  with  complex  terms. 

m  +  n 

o7     ,   -  r-  1  2a;  ,     3v 

3bc-}-;jXT/—7mn  and h 


a       a-\-l 
trinomials  :  the  first  is  entire,  the  second  fractional. 


18  PKIMARY  DEPEtTlTIONS  AKD   SIGNS.  [I. 

So,    y/a±lopl  —  V<^    ^^^    ^PQ  —  ^PQ  +  PQ  —iPQ    are 
quadrinomials,  but  reducible  to  monomials;  viz.,  to 

0  and  2}rpq\ 

The  above  examples  are  literal ;  the  following  are  numerical : 

V^  ^-V'^*   2   +3%  1  +  V— 1  are  binomial  surds. 

1  +  -^2  —  -{/—  3  is  a  trinomial  surd. 

1.1  X  1.01  X  1.001  X  •••  is  an  infinitel}^  continued  product. 

is  an  infinite  continued  fraction,  but  one 

2  4-1 

~ whose  value  is  V^  —  1 ,  an  irrational 

~  ■ finite  number,  as  will  appear  later. 

An  expression  may  be  entire,  rational,  etc.,  as  to  some  of  its 
letters  onl}-. 

E.g.,   ^'^y  ^  is  rational  as  to  a,  m  and  n, 
m-f-n 
and  it  is  entu*e  as  to  a ; 

but  it  is  irrational  as  to  &  and  c, 

and  it  is  fractional  as  to  m  and  n. 

"When  the  terms  of  an  expression  are  so  related  to  each  other 
that  each  successive  term  is  derivable  by  some  fixed  law  from 
the  previous  terms,  the  expression  is  a  series. 

E.g.,    l-f-a;-f-»^-fic^H \-^  is  a  finite  series  if  r  is  any 

given  integer ; 

but  l  +  a;-fic^4-a^-}--"  +  afH is  an  infinite  series. 

.  In  this  series  of  is  called  the  general  term,  because  by  giving  to 
r  in  turn  the  values  0,  1,  2,  3,  . . . ,  or  any  of  them,  all  the  terms 
of  the  series,  or  any  of  them,  are  found. 

When  the  values  of  the  several  letters  in  a  literal  expression 
ire  known,  then  the  value  of  the  expression  may  be  found  by 
substituting  these  values  in  place  of  the  letters,  and  performing 
the  operations  indicated. 

E.g.,    if  a  =  2,    5  =  3,    c  =  4, 
then         a6c  =  24,    a  +  6  — c=l,    a  :  (6 -f  c)  =  |^. 

So,       if  a;  ==  a  +  &     and     y  =  a  —  h, 
then         x-{-y—2a,     x  —  y  =  2b,     xy  =  a^  —  h^. 


§  13.]  FUNCTIONS.  19 

A  literal  expression  ma}^  be  entire,  fractional,  rational,  etc., 
but  its  numerical  value  not  so  ;  or  the  reverse. 

E.g.^  a;  is  entire,  o;"^ fractional,  -y^o?  irrational,  ?/^ rational; 
but,  if  aj  =  ^  and  y  —  -^2, 

then         the  value  of  x  is  fractional,  that  of  x~^  is  entire,  that 
of  -^/x  is  rational,  and  that  of  'if  is  irrational. 
Manifestly,  if  all  the  letters  stand  for  integers  and  the  expres- 
sion is  entire,  its  value  is  an  integer. 

E.g.^    if  a  and  h  are  integers,  (a  —  6)  {2o?  -\-  36^)  is  integral. 
As  to  an}'  of  its  letters,  an  expression  is  symmetric  when  its 
value  remains  unchanged  however  those  letters  exchange  places. 
E.g.^   xyz  and  x-{-y  +  z  are  symmetric  as  to  a;,  y,  and  2;,  or  as 
to  any  two  of  them. 
So,  w-\-x  —  y  —  ziQ  s^-mmetric  as  to  w  and  a;,  and  as  to  y 

and  z ;  but  not  as  to  w  and  2/,  to  w  and  2;,  to  x  and 
2/,  nor  to  x  and  z. 
An  expression  is  converted  or  transformed^  when  changed  in 
form  but  not  in  value  ;  developed  or  expanded^  when  transformed 
into  a  series. 

§  13.     FUNCTIONS. 

If  a  number  is  so  related  to  other  numbers  that  its  value 
depends  upon  'their  values,  it  is  a  function  of  those  numbers : 

an  {  '.''P^'f.^  function,  Tvhen  {  •^^f^^^^l      ,  in  terms  of  those 
>  implicit  '  '  not  expressed 

numbers.     The  numbers  are  the  arguments  of  the  function. 

E.g.,    m  u  =  Sxy,  u  is  an  explicit  function  of  the  arguments 
X  and  y ; 
but  X  is  an  implicit  function  of  the  arguments  u  and  y, 

and  y  is  an  implicit  function  of  the  arguments  u  and  x. 

So,       in  y^  =  u:3x,  y  is  an  implicit  function  of  u  and  x ; 
but  in  y  =^{u:Sx),  y  is  an  explicit  function. 

An  explicit  function  of  one  or  more  numbers  is  known  (given 
or  determined)  in  terms  of  those  numbers.  It  is  symmetric, 
algebraic,  transcendental,  rational,  etc.,  according  as  the  expres- 
sion which  gives  its  value  is  symmetric,  algebraic,  etc. 


20  PEIMABY  DEFINITIONS  AND   SIGNS.  [I. 

If  one  number  (function)  depends  upon  its  arguments  in  the 
same  way  as  another  number  depends  upon  its  own  arguments, 
i.e.,  if  the  expressions  involved  are  of  the  same  form,  then  the 
first  number  is  the  same  function  of  its  arguments,  as  the  second 
number  is  of  its  arguments. 

E.g.^    if  x^-{-x  =  a    and    y'-\-y  =  b^ 

then      ^  "  is  the  same^  f^^^\\^'}_  function  of  ^  ''^  as  |  ^  is  of  ^  f 
'  X  '  imphcit  'a        '  ?/  '6. 

So,       the  expression  x-\-2y  is  the  same  function  of  x  and  y 

as  a  -f  2  6  is  of  a  and  &,  and  the  same  as  ?/  +  2  a;  is 

of  y  and  x. 

A  function  may  ])e  denoted  by  the  letters/,  f,  <^,  ... ,  with  or 

without  indices,  and  followed  by  the  arguments,  either  enclosed 

in  a  parenthesis  or  not. 

rvr.  /j««^+«  J  t^e  same    ^„„„<..  „  j  the  same  letter 

To  denote  {  ^  ^.^^^^^^  function  ^  ^  ^.^^^^^^  ^^^^^^  ^^  .^^^^  is 

used. 

E.g.,iff(x)  =  x^  —  ax, 
then         /(?/)  =7f  —  ay  during  the  same  investigation  ; 
but  f{y)  cannot  stand  for  a^  —  ay^  nor  for  ay  —  if. 

So,  if  F(a;,  a)  =a^  —  ax, 
then         F{y^  b)  =y^  —  by,     F(a,x)~a^—xa,  .... 

But  if  F'(a;,  a)  =  ar'  —  a^x,  or  any  other  form, 
then         f'(6,  y)  =  b^  —  y^b,   the  same  form. 

If         F(a;,2/)  J  F(^,a;), 

then         either  is  ^  ^  symmetric        function  of  a;  and  2^. 
•  an  uns^Tnmetric  ^ 

E.g.,  if  f(»,  y)  =/(x)  -/(i/) ,   or  =f(x^j) , 
then         F  denotes  a  S3'mmetric  function  ; 
but  not  if  F(ic,  y)  =f{x)  -/'{y) ,   or  ~f{x :  y) . 

So,       ^{x,y)-\-<fi{y,x),   butnot  </)(a;,2/)  — <^(?/,  a;),    is  sjtu- 
metric. 

So,  if  F(a;,2/,2),   F(x,z,y),  Y{y,z,x),  F(y,x,z),  F(z,x,y), 

all 
and  f(z,  y,  x)  be^  ^^^  ^^^  identical, 

4.,  .,,       .    ,  a  symmetric         ^      ^'        ^ 

then         either  is  -^       -^  ,  .    function  of  x.y.z. 

'  an  unsymmetric  '  ^ ' 


§§  14,  15.]  COErnCIENTS.  —  DEGREE.  21 

§  14.     COEFFICIENTS.  —  LIKE  AND  UNLIKE  TERMS. 

When  a  number  is  the  product  of  several  factors,-  they  are  its 
co-factors;  and  anj-  one  of  them,  or  the  product  of  am'  two  or 
more  of  them,  is  a  coefficient  of  the  product  of  the  remaining 
co-factors.  A  coefficient  is  numerical^  literal,  or  mixed^  accord- 
ing as  it  is  a  numeral,  a  letter  or  letters,  or  a  numeral  and  let- 
ters combined. 

E.g.^    in  7a6c,  7  is  the  coefficient  of  a&c,  7a  of  6c,  lab  of  c, 

76  of  ac,  7  c  of  a6,  76c  of  a, 

Usuall}'  the  numeral  alone,  together  with  the  sign  of  the  num- 
ber, -f-  or  — ,  is  counted  as  the  coefficient. 

Terms  which  differ  only  in  their  coefficients  are  like  (similar) 
terms  ;  other  terms  are  unlike. 

E.g.^    5 ax  and  lax  are  like,     but  6 ax  and  76?^  are  unlike. 
So,       5  ax  and  7  bx  are  like  if  5  a  and  7  6  are  counted  as  the 
coefficients  of  a; ;  but  unlike  if  5  and  7  be  coefficients 
of  ax  and  bx. 
So,       3V(«'+&'),  5a^{a^+b'),  {7b +  dc)^{a'-hb')  are 

like  surds. 
But      3  V(«'+  ^') ,  5 a ^(a2  +  62) ,  (7 5  +  9 c)  ^(a^  +  c')  are 
all  unlike  surds. 

§  15.     DEGREE. 

The  sum  of  the  exponents  in  a  simple  term  is  its  degree.  The 
degree  of  a  polynomial  is  that  of  the  term  whose  degree  is  high- 
est of  all.  A  polynomial  made  up  of  simple  terms  all  of  the 
same  degree  is  homogeneous.  Expressions  having  the  same  de- 
gree are  homogeneous  with  each  other. 

E.g. ,    a^  H-  3  a^6  +  3  a6^  +  6^  is  homogeneous,  of  the  3d  degree. 
So,       a",  a"-^6,  a«-26S   ...,  a^'-^'b^  ...,  a6^-S  6"  are  homo- 
geneous with  each  other  and  of  the  nth  degree. 
and  ax^,  b^xy,  (?]f'  are  of  the  2d  degree  and  homogeneous 

with  each  other  as  to  x  and  y  ; 
but  of  the  3d,  4th,  and  5th  degrees  respectively,  and  not 

homogeneous,  as  to  all  the  letters. 


22  PEIMARY  DEFINITIONS  AND   SIGNS.  [I. 

So,  the  trinomial  a~^  +  b~^  +  c~^  is  of  the  —1st  degree,  and 
not  homogeneous ; 

and  a^  +  6^  +  c^  is  of  the  3d  degree  and  not  homogeneous ; 

and  ma?h~''ar  +  vra^b'^xy  -{-]fa^h~^'if  is  homogeneous  and 

of  the  2d  degree  as  to  x  and  y^  and  homogeneous 
and  of  the  —  3d  degree  as  to  a,  6,  a;,  and  y ; 

but  it  is  not  homogeneous  if  m,  w,  and  p  be  also  counted, 

for  then  the  first  term  is  of  the  —2d  degree,  the 
second  term  is  of  the  —1st  degree,  and  the  last 


term  is  of  the  0th  degree. 


So,       the  binomials  ^  -  M'  and  ^a"  -  fs -^b'-  i  (^  b 
b       5a^  \  \^J 


are 


homogeneous  and  respectively  of  the  1st  and  the  -f 

degree. 
The  degree  of  a  product  is  the  sum  of  the  degrees  of  the  fac- 
tors. The  degree  of  an}-  power  of  an  expression  is  the  product 
of  the  degree  of  the  expression  b}'  the  exponent  of  the  power. 
The  degree  of  a  quotient  is  the  excess  of  the  degree  of  the 
dividend  above  the  degree  of  the  divisor.  A  power  or  product 
or  quotient  of  homogeneous  expressions  is  homogeneous,  and  a 
sum  of  homogeneous  expressions  of  any  same  degree  is  homo- 
geneous and  of  that  degree. 

E.g.,    {a--^b')^'(ay^-hy^)^:  (ab  +  xy)^  is  of  the  1st  degree 
and  homogeneous  as  to  all  the  letters, 
but  it  is  of  the  0th  degree  and  not  homogeneous  as  to  a  and 

b  only,  or  as  to  x  and  y  only. 


§  16.]  EXAMPLES.  23 

§  16.     EXAMPLES. 

1.  In  the  sentence   (cc  +  a)^— (ic— a)^  =  4aa;,   point  out  the 

verb,  nouns,  conjunctions,  and  phrases,  and  state  their 
grammatical  relations. 

§2. 

2.  Translate  and  read  in  words  the  following  symbols : 

3.  Write  in  symbols  : 

p  sub  naught,  q  second,  x  prime  sub  r  prime,  large  x 
fourth  sub  a  prime,  large /sub  i  third  and  sub  k. 
§3. 

4.  If  a  =  2  and  6  =  —  3,  which  of  the  following  numbers  are 

positive,  and  which  negative  ? 
a,     6,     —a,     —6,     2a,     5&,     —8a,     — 11&. 
§5. 

5.  Connect  each  of   the   following  pairs  of  numbers  by  the 

appropriate  sign  >  or  <  ;  also  by  the  sign  >  or  <  : 
0,1;  0,-1;   -1,0;    2,1;    1,-2;    -2,-1;   -1,-2; 
~x^~2x;  '^x,~2x;  ~a-\-~b,~a  —  ~b. 

6.  Read  in  words  the  statements  : 

If  a  <  6  ^  0,   then  a  ^  6  ; 

If  a  >  6,  ^  0,   then  a  ^  6  ; 

and  explain  the  meaning  of  the  copulas  used  therein. 

7.  Correct  the  following  continued  statements  by  introducing 

or  suppressing  commas : 

3<— 4,<1>1;       '.'x^a,     .-.  3a7>  3a,  >  2a. 
§§  6,  7. 

8.  Read  in  words  the  following  formulae  : 

-a+-6  =  -(+a4-'^&)  ;    +a--5>0;    -a-+&<0. 

9.  Correct  the  following  statements  by  introducing  the  proper 

brackets  : 

5-3  +  1  =  1;    5-3-1  =  3;    -5+4-l=-8. 


24  PRIMARY   DEFINITIONS   AND   SIGNS.  [I. 

10.  Read  in  words  the  statements  : 

(a  +  6)+(a-6)  =  2a;    (a  +  b)-{a-b)  =  2b -, 

and,  considering  a  and  b  to  stand  for  any  two  numbers 

whatever,  read  these  two  statements  as  general  truths. 

§§  8,  9. 

11.  Separate  the  portions  of  the  following  continued  statements 

where  necessary  to  avoid  false  equations  or  inequalities : 
2x3  =  6  +  4=10-f-5  =  2; 

12.  Read  and  verify  the  statements  : 

1  ! .  2  ! .  3  ! .  4^  =  1*.  2^.  3-.  4^  3  !  !,=  (3  !)  !,=  (3  !)-•  4  •  5. 

13.  Correct  the  following  statements  by  introducing  the  proper 

brackets : 

30-!-3xo  =  2;    30-^10-^5  =  15;    5a;-4a;x  l+2  =  3a;. 

§  10. 

14.  Translate  into  words  :  , 


a^-\-ab 
+  bc 
-\-ca 


X  +  abc ; 


[(a  +  6).c-(a;-2/)T-S[(«-^)-«]'-(^-2/)"S~'- 

15.  Interpret  the  following  expressions  and  statements : 

2^;  J;   (2a;)-3;   (G?/)"' ;  8^  =  4;  2<*  =  3«  =  4«=  .... 

16.  Introduce  brackets  so  that  2^  shall  equal  64  ;  256. 

17.  Whatpowerof  ojis  [(a^)2]2?    (a^)^?    (a^'y?    a;(3V?    of"? 

§  11. 

18.  Find  the  value  of  : 

log28,      logai,       log22,      logsl,     log22^2,     log^8, 
logii,     log82,      logsj,     log^2,    log^i,  log4l6. 

19.  Of  what  number  ia  4  the  logarithm  to  base  2  ?  to  base  4  ?  to 

base  ^?  to  base  -I-? 

20.  To  the  base  10,  of  what  number  is  3  the  logarithm?  2 ?   1  ? 

0?  -1?   -2?  f?  I?  -i?   -f? 

21.  To  what  base  is  2  the  logarithm  of  9  ?   of  27  ?   of  i? 

22.  To  what  base  is  i  the  logarithm  of  5?  of  yo?  of  i? 


§16.]  EXAMPLES.  25 

§  12. 

23.  If  a=l,  &  =  -3,  c  =  5,  find  the  value  of : 

a2&2^1       i-aV      26'-4ac      a^4-2a5-jr&^ 

24.  If  a  =  25,  6  =  9,  c  =  — 4,  d  =  — 1,  find  the  values  of : 

Va'-2-^6^  +  3-^c^-4^^d^; 

V-  &c  +  3  VacfZ  -  4  V-  &'^  +  V-c^<^- 

25.  If  a  =  0,  6  =  —  2,  c  =  4,  d  =  — 6,  find  the  value  of : 

26.  If  a  =  2,  a;=16,  find  the  values  of : 

log„a;,  log^V^'  log«i»^,  log„(log„a;),  log«[log41og„«)], 
log,a,  log,Va.  log.a'»  log,(log,a),  log,[-log,(log,a)]. 

27.  In  Ex.  23-26  show  which  expressions  are  algebraic,  which 

transcendental,  which  entire,  which  fractional,  which  ra- 
tional, which  irrational:  first  in  form,  second  in  value. 
Show  what  portions  of  them  are  symmetric,  and  as  to 
which  letters. 

§  13. 

28.  If  <jf)(a;)  =  a:^  +  3  a;  +  6,  write  the  expressions  for : 

<f>{a),    cf>{2a),    cf>{-y),    </)(a;  +  2/)»    <l>(^-y), 
and  find  the  values  of    «^  (0) ,    <^  (1) ,    </>  (  —  2) . 

29.  If  cfi{x,  y,z,t)  =  a^-\-Syz-{-  f,  write  the  expressions  for : 

<^(a2,w,7i,Z),    <^(0,l,-2,  cc),   </>(a;,a;,ic,cc),    <f>(t,x,y,z). 

30.  As  to  what  letters,  if  any,  is  each  of  the  following  func- 

tions symmetric? 

<f>{x+7j)',     F{xy,x-\ry);    f{x,y,z)-\-f{y,z,x)+f(z,x,y)  ; 

<f,(x)+cf>(y) -{-(}>' (z)  ;     F{x,yz)-\-F(y,zx)-\-F(z,xy). 

§  14. 

31.  Show  what  factors  must  be  taken  as  coeflScients  in  order 

that  the  following  sets  of  terms  shall  be  like  : 
3a6,  36c;      5axr,  2axy;      Imn^^Am^n; 
2a6c,  36ccZ,  Acdx'^      i^V^i  iV^Vi^'^y^)- 
§  15. 

32.  In  Ex.  23-25  state  the  degree  of  each  one  of  the  expres- 

sions, and  show  which  of  them  are  homogeneous. 

OF  THE     "^ 


26  PRIMARY  OPERATIONS.  [II. 

II.    PRIMARY  OPERATIONS. 

§  1.     LOGICAL    TERMS. 

A  DEFmrriON  is  a  statement  of  the  sense  in  which  a  word  or 
S3Tnbol  is  used. 

A  theorem  is  a  general  truth:  if  self-evident,  it  is  an  axiom; 
if  auxiliarj-  to  a  following  theorem,  it  is  a  lemma;  if  an  obvious 
consequence  of  a  previous  theorem,  it  is  a  corollary. 

A  theorem  consists  of  two  parts,  the  hypothesis  or  data,  and 
the  conclusion  which,  if  not  self-evident,  is  to  be  established  by 
a  demonstration. 

A  converse  of  a  theorem  is  another  theorem  that  has  for  data 
the  conclusion,  or  the  conclusion  and  an}-  of  the  data,  of  the  first 
theorem,  and  for  conclusion  some  datum  of  the  first  theorem. 

E.g.,  the  theorem  "  If  from  equal  numbers  equals  be  subtracted, 
the  remainders  are  equal,"  is  an  axiom,  wherein  the  clause  before 
the  comma  is  the  h3'pothesis,  and  the  clause  after  the  comma  is 
the  conclusion.  It  needs  no  demonstration.  Its  converses  are  : 
"'  K  the  remainders  be  equal,  the  numbers  from  which  equals  are 
subtracted  are  equal,"  and  ''If  the  remainders  be  equal,  the 
numbers  subtracted  from  equals  are  equal." 

Of  demonstrations  three  kinds  are  found  in  Algebra : 

(a)  Direct  proof,  wherein  the  conclusion  follows  as  a  direct 
and  necessary  consequence  of  certain  axioms  and  definitions, 
and  of  other  theorems  already  proved. 

(b)  Proof  by  exclusion,  also  called  reductio  ad  absurdum,  or 
indirect  proof  wherein  are  first  enumerated  all  possible  conclu- 
sions from  the  given  data,  and  then  the  ti*uth  of  one  of  them  is 
established  bj'  the  exclusion  as  absurd  of  all  the  rest. 

(c)  Proof  by  induction,  which  consists  of  three  steps  : 

1 .  Proof,  either  direct  or  indirect,  that  the  theorem  is  true 
when  applied  to  one  or  more  cases  at  the  beginning  of  a  series 
of  particular  cases  of  the  general  theorem. 


§  2.]         C03IBIXAT0RY  PROPERTIES   OF  OPERATIONS.  27 

2.  Proof  that,  if  the  theorem  be  true  up  to  any  one  case  in- 
clusive, then  it  must  also  be  true  for  the  next  higher  case  in  the 
series. 

3 .  Proof  by  progressive  steps  that,  since,  beginning  with  the 
cases  actually  proved  (1),  it  is  true  for  the  next,  and  the  next, 
and  the  next,  indefinitely  (2) ,  therefore  it  is  universally  true. 

A  problem  is  anj^thing  to  be  done  ;  usually,  in  Algebra,  it  is  to 
find  numbers  or  expressions  that  will  satisfy  given  conditions. 
These  numbers  or  expressions,  together  with  the  process  of  find- 
ing them,  constitute  the  solution  of  the  problem. 

A  solution  is  -(  ^     , .    ,     when  it  gives  •{  '         of  the  numbers, 
'  particular  °         '  some  * 

or  expressions,  or  sets  of  numbers  or  expressions,  that  satisfy 
the  given  conditions.  Usually  the  general  solution  is  sought,  with 
a  demonstration  showing,  by  previous  theorems  and  problems, 
that  the  solution  satisfies  the  given  conditions  and  is  general. 

A  cheeky  or  test^  is  a  comparison  of  results  designed  to  detect 
any  accidental  errors  in  the  work. 

A  postulate  assumes  as  self-evident  that  the  solution  of  a 
problem  is  possible. 

The  letters^  «-^-"-  at  the  end  of  a^  demonstration  ^  ^^^  ^ 
'  Q.E.F.  '  solution 

^     ■,        ,1  demonstrandum         ,  .  ,  +    i,    j  proved. 

quod  erat  \  ^^,,.,^^^^  -  which  was  to  be  {  {^^^^^ 

§  2.     COMBINATORY  PROPERTIES  OF  OPERATIONS. 

An  Algebraic  Operation  is  an  act  by  which  two  or  more 
numbers,  the  elements^  are  combined  together  to  produce  one 
number,  the  result. 

Manifestly,  the  result  is  a  function  of  the  elements. 

An  operation  is  ^  ^^^^^f     when  {  ^^^^'  , ,       two  elements  are 
^  '  complex  '  more  than 

combined.  If  a  complex  operation  consist  of  two  or  more  simple 
operations,  and  if  they  be  all  of  the  same  kind,  it  is  a  continued 
operation. 

E.g.^  the  continued  addition  of  three  numbers  consists  of  first 
adding  two  of  them,  and  then  adding  the  third  number  to  this  sum. 


28  PEtMARY   OPERATIONS.  [11. 

Of  the  two  elements  of  a  simple  operation,  one,  the  operand, 
is  conceived  of  as  acted  upon  by  the  other,  the  operator,  in  a 
way  shown  by  the  sign  of  operation. 

E.g.,    m  6  +  2  =  8,    6-2  =  4,    6x2  =  12,    6:2  =  3, 
the  operand  is  6  ;  the  operator  is  2  ;  the  results  are  8,  4,  12,  3  ; 
the  operations  are  addition,  subtraction,  multiplication,  division ; 
and  the  signs  of  operation  are  +,  — ,  X  ,   : . 

So,       in  16^  =  256,    -^16  or  V16  =  ±  4,   log2l6  =  4, 
the  operand  is  16  ;  the  operator  is  2  ;  the  results  are  256,  ±4,  4; 
the  operations  are  involution,  evolution,  the  finding  of  a  loga- 
rithm ;  and  the  signs  of  operation  are,  the  position  of  the  expo- 
nent, -y/,  the  word  ''log." 

I  uni-determinate, 
An  operation  is  <  multi'determinate,  when,  from  given  elements, 
'  indeterminate, 
I  only  one  result, 
it  gives  <  several  different  results,  but  none  intermediate. 

'  an  infinite  number  of  results  in  a  continuous  series. 

The  rational  operations  (addition,  subtraction,  multiplication, 
division,  and  involution  to  integral  powers)  and  the  finding  of 
logarithms  are  generally  unideterminate  ;  but  evolution  is  gener- 
all}'  multideterminate ;  and  operations  with  special  elements  are 
often  indeterminate. 

E.g.,    6+2,  =8;    6  -  2,  =  4 ;    6  X  2,  =  12  ;    6  :  2,  =  3  ; 
3^,=  9  ;    log 3  9,  =  2  ;  are  unideterminate  ; 
but  -y/9,  =  either  +  3  or  —  3,  is  multideterminate  ; 

and  0  :  0,  0^  logoO,  logjl,  are  indeterminate. 

men  the  result  and  the^  °P-^,^  are  given,  the^  "P-^^^ 

may  be  found  by  an  operation  called  the  {  "^       -.  inverse  of  the 

original  or  direct  operation, 

wherein  the  operand,  operator,  and  result 

-the      ^-5      :Pi^M:^:S-P-«vely,ofthe 

direct  operation.  Hence  an  inverse  operation  is  the  undoing  of 
what  was  done  by  the  direct  operation,  and  it  ends  where  the 
direct  operation  began. 


§  2.]         COMBINATORY  PKOPERTIES  OP  OPERATIONS.  29 

An  inverse  operation  may  be  defined  as  an  operation  "the 
efiect  of  which  the  direct  operation  simply  annuls  "     It  consists 
not  in  any  new  procedure,  "  but  in  a  series  of  guesses  suggested 
b}'  prior  general  knowledge  of  the  results  of  the  direct  operation, 
and  tested  by  the  direct  operation  itself."  —  Boole, 
E.g.,    6-2=     4     •.•     4  +  2  =  6; 
6  :  2=      3     •.•     3  X2  =  6; 
V9     =±3     •••     (+3)2  =  9  and   (-3)2  =  9; 
log39=      2     •••     32       =9; 
An  inverse  operation  is,  therefore,  described  b}"  the  two  words 
"guess"  and  "test."    The  error  of  one  guess  helps  the  next  one. 
E.g.,   To  divide  756  by  27  : 

Guess  30  ;  that  is  too  large,  for  the  product,  27  x  30, 

is  810,  which  is  larger  than  756. 
Guess  20  ;  that  is  too  small,  for  the  product,  27  X  20, 
is  540,  and  the  remainder,  216,  is  larger  than  27. 
Guess  8  as  the  quotient  of  the  remainder  216 ;  27 ; 
this  guess  is  right,  for  the  product,  27x8,  is  216; 
and  Uie  whole  quotient  is  28,  the  sum  of  20  and  8. 
An  inverse  operation  may  or  may  not  be  multideterminate 
when  the  direct  operation  is  unideterminate ;   and  the  two  in- 
verses may  or  may  not  be  of  the  same  kind. 

E.g.,    Direct  Operations.         First  Inverses.        Second  Inverses. 
6  +  2=    8,  8-2=      6,  8-6  =  2; 

6x2  =  12,  12  ;  2=      6,         12   :  6  =  2  ; 

62       =36,  V36     =±6,         log636  =  2; 

wherein  the  two  inverses  of  addition  are  both  subtraction  and 

unideterminate ; 

and  of  multiplication,  they  are  both  division  and  unideterminate  ; 

but  of  involution,  the  first  is  evolution  and  multideterminate,  and 

the  second  is  the  finding  of  a  logarithm  and  unideterminate. 

A  direct  simple  operation  is  sometimes  the  repetition  of  more 

elementary  operations. 

E.g.,  addition  of  a  ^  ^  ^  . ^      integer,  ±  m,  is  counting  a 
unit  {  Q^  fn>  times  ;  and  addition  of  a-j  ^eo-ath^e  ^^^^^i^^'  ^  ■^' 


30  PKIMARY  OPEEATIONS.  [U. 

is  counting  m  times  -{  ^  such  a  number  as,  if  counted  on  n  times, 
would  add  a  unit. 

So,  multiplication  bya^  P^^^J'^^Mnteger,  ±  m,  is -J  ^tLin^ 

the  multiplicand  m  times  -{  ^.        0 ;    and  multiplication  by  a 

{  positive  f      ;  m   j^  adding  ;        _,  to  ^ 

^  negative  '      n'       »  subtractmg  '  from 

a  number  as,  if  added  n  times  to  0,  would  give  the  multiplicand. 

So,  involution  by  a  -{  ^  ^  . .  ^  integral  exponent,  ±  m,  is 
•{  ^".  |P  ^^°^  1  by  the  base  m  times ;    and  involution  b}^  a 

<  Stive  fractional  exponent,  ±  f ,  is  ^  Svidin^^  ^^  ^  ^^^^^' 
by  such  a  number  as,  if  multiplied  n  times  into  1,  would  give 
the  base. 

Thus  the  operations  of  addition,  multiplication,  and  involution 
all  come  from  the  more  elementary  operation  of  counting. 

So,  often,  when  the  operator  is  a-[  ^p^!,Ji^^g  integer,  as  ±  m, 
then  the  more  elementary-}  .  ^  operation  is  performed  m 
times  in  succession  upon  the  operand  ;  and  when  the  operator  is 
a-{  P  ^^  !^f  fraction,  as  ±  — ,  then  some  operation  is  performed 
m  times  which,  if  performed  n  times,  would  be  equivalent  to 

the  more  elementary -j  .  operation. 

•^  '  inverse    ^ 

The  modulus  of  a  simple  operation  is  that  operator,  if  any, 
which  always  makes  the  result  equal  to  the  operand. 

E.g.^  •.'  x-\-0  =  x    and    a;  —  0  =  a;,  [x  any  number 

a;  X  1  =  «     and     x  :   1  =  a;, 
and  3^       =x     and     -y/x     =a;; 

.'.  the"  modulus  of  addition  and  subtraction  is  0, 
the  modulus  of  multiplication  and  division  is  1, 
and  the  modulus  of  involution  and  evolution  is  1. 


§2.]        COJ^IBINATORY  PEOPEKTIES   OF  OPERATIONS.  31 

.  ....  commutative  a      ^^      ^         j.    \  can 

An  operation  is  <  _^^,  ^^__„  +^.  v.^  when  the  elements  < 

^  »  non-commutative  »  cannot 

exchange  places  without  changing  the  result ;  i.e.,  when  the  result 

.    I  a  symmetric        4^      i.-        -^  ^i,      i  *. 

is  <       -^  .  .    function  of  the  elements. 

'  an  uns3'mmetric 

E.g.^    Commutative  Operations.  Non-commutative  Operations. 

2  +  3  =  3  +  2,  ,    2-3=?b3-2, 

2x3  =  3x2.  2:3:?t3:2, 

23        ^Z\ 
-^2     =?b-^3, 

I0g32:?t:l0g23. 

associative 
A  continued  operation  is  \  „o„.„,,o,j-„j,-„,  when,  as  long  as  tlie 

elements  do  not  exchange  places,  they-{  ^^^^     .  be  grouped  at 
will  without  changing  the  result. 

E.g.^    Associative  Operations.  Non-associative  Operations. 

(12+4)  +  2  =  12  +  (4  +  2),     (12-4)- 2:^  12 -(4-2), 
(12x4)  X  2  =  12  X  (4x2),      (12  :  4)  :  2  :^  12  :   (4  :  2), 

(12*)2  =5^12(4'). 

A  second  operation  is^  nZdMhutwe  ^'  ^^  ^  ^'"^  operation 
when  the  final  result  •{  |^  ,  the  same,  whether  the  second  opera- 
tor act  upon  the  result  of  the  first  operation,  or  upon  the  separate 
elements  of  the  first  operation,  and  then  these  results  are  com- 
bined by  the  first  operation.  An  operation  distributive  as  to 
addition  is  also  called  simply  distributive,  or  linear. 

E.g.,        Distributive  Operations.  Non-distributive  Operations. 

Multiplication  as  to  addition.  Addition  as  to  multiplication. 

12  +  6  X  3  =  12x3 +Gx3  ;       12x6+3  =7^  12+3  X  6"+3. 
Involution  as  to  multiplication.  Involution  as  to  addition. 


12X6        =12^x6^;  12+6        =7^122+6^; 

Evolution  as  to  multiplication.  Evolution  as  to  addition. 

-v/27x8    =^27x-^8;  a/27±8   =^-^27  ±^8; 

Finding  of  logarithms  as  to  addition. 

log6(216  +  36)  =^  log6216  +  log636 
Finding  of  logarithms  as  to  multiplication. 

log6(216  X  36)  =^  log6216  X  log636. 


32  PBIMAKY  OPERATIONS.  [II. 

§  3.     AXIOMS. 

1 .  Numbers  equal  to  the  same  number  are  equal  to  each  other. 

2.  Kto  equal  numbers  equals  be  added,  the  sums  are  equal. 

3.  K  from  equal  numbers  equals  be  subtracted,  the  remain- 
ders are  equal. 

4.  If  equal  numbers  be  multiplied  by  equals,  the  products 
are  equal. 

5.  If  equal  numbers  be  divided  by  equals,  the  quotients  are 
equal. 

G.  K  equal  numbers  be  raised  to  like  integral  powers,  the 
POWERS  are  equal. 

7.  K  of  two  equal  numbers  like  roots  be  taken,  ever}^  root 
of  the  first  number  is  equal  to  some  root  of  the  other. 


8.  If  of  three  numbers  the  first  be-j  ?  than  the  second, 

and  the  second  be  equal  to  or  -{  W^         than  the  third,  then  is 
the  first  {  ^^^^^  than  the  third. 

9.  If  one  number  be  -{  F^  than  another,  and  if  to  each  of 
them  be  added  the  same  number  or  equal  numbers,  then  is  the 
first  sum^  \qss^^^  ^^^^  ^^®  ^*^^^- 

10.  If  one  number  be  <[  ¥^^^  ^^  than  another,  and  if  from  each 
of  them  be  subtracted  the  same  number  or  equal  numbers,  then 
is  the  first  remainder  -{  y^  ^^  than  the  other. 

11.  If  one  number  be-{  P*^^  ^  than  another,  and  if  each  of 
them  be  subtracted  from  the  same  number  or  from  equal  num- 
bers, then  is  the  first  remainder  ■{  .      than  the  other. 

12.  If  one  set  of  numbers  be -J  ¥^^^  ^^  than  another  set  of  as 

many  more,  each  than  each,  then  is  the  sum  of  the  first  set  -J  ^ 
than  the  sum  of  the  others. 


§3.]  AXIOMS.  83 

13.  If  one  number  be-{  F[f^  ^^  than  another,  and  if  each  of 
them  be  multiplied  or  divided  by  the  same  or  equal  positive  num- 
bers, then  is  the  first  product  or  quotient  -{  ^  than  the  other. 

14.  If  one  number  be-{  P  than  another,  and  if  each  of 

them  be  multiplied  or  divided  b}'  the  same  or  equal  negative 

less 
numbers,  then  is  the  first  product  or  quotient-;         ,      than  the 

other. 


15.  If  of  three  numbers  the  first  be  {  ^^^^^  than  the  second, 
and  the  second  be  equal  to  or  -{       ^-.^      than  the  third,  then  is 

the  firsts  ^^^l^^^  than  the  thu-d. 

16.  If  one  number  be  ■{  *=,,  .  than  another,  and  if  each  of 
them  be  multiplied  by  or  into  the  same  number  or  equal  num- 
bers, then  is  the  first  product^  smaller  *^^^  *^®  other. 

17.  If  one  number  be -{  ^..  than  another,  and  if  each  of 
them  be  divided  by  the  same  number  or  by  equal  numbers,  then 
is  the  first  quotient  ^    aiger    tj^an  u^g  Q^jigP^ 

18.  Kone  number  be-j  ^^»^-^  than  another,  and  if  the  same 
number  or  equal  numbers  be  divided  b}^  each  of  them,  then  is 
the  first  quotient^  Wer^  *^^^  *^®  ^*^^^- 

19.  If  one  set  of  numbers  be  -{  ^^^^^  than  another  set  of  as 
many  more,  each  than  each,  then  is  the  product  of  the  first  set 
-J  gjvj^iig,.  than  the  product  of  the  others. 

20.  If  one  number  be -{  ^^^^^  than  another,  and  if  like  posi- 
tive powers  or  roots  of  them  be  taken,  then  is  the  first  power  or 


34  PRIMAEY  OPEKATIONS.  [II.  th. 

21.  If  one  number  be -J  ^|,  than  another,  and  if  like  nega- 
tive powers  or  roots  of  them  be  taken,  then  is  the  fii'st  power 
or  root  -{  ,    ^       than  the  other. 

22.  If  two  numbers  be  opposites,  one  of  them  is  positive 
and  the  other  is  negative  ;  they  are  equall}-  large  ;  and  their  sum 
is  naught. 

23.  If  all  the  letters  of  an  entire  expression  stand  for  integers, 
the  value  of  the  expression  is  an  integer. 

Note  1.  For  convenience,  and  because  quite  evident,  all  the 
propositions  above  given  are  called  axioms,  although,  in  strict- 
ness, some  of  them  are  deducible  from  others. 

E.g.^   Ax.  1  is  deducible  from  Ax.  8. 

For      let  A,  B,  c  be  three  numbers  such  that  a  =  c  and  b  =  c  ; 
then         either  a  =  b,    orA>B,    orA<B;  and  of  these  three, 
the  onl}'  possible  conclusions  from  the  data,  one 
must  be  true,  and  the  others  false. 

Suppose  A  >  B  ; 
then    •.•A>B     and    b  =  c,  Di3T' 

.*.  A>c,  [ax.  8 

a  consequence  from  the  supposition  a  >  b,  which  is  contrar}^  to 
the  hypothesis  of  the  theorem,  and  therefore  absurd. 

.*.  the  supposition  a  >  b,  which  led  to  this  absurd  conse- 
quence, is  itself  absurd  ;  and  a  >  b. 

So        it  ma}'  be  proved  that  a  <  b  ; 
and     •.•a>b     and    a<b, 

.•.  it  is  onh'  left  that  a  =  b.  q.  e.  d. 

Note  2.  The  reader  will  observe  that  the  form  of  statement 
'.s  different  in  Ax.  7  from  that  in  any  of  the  others.  It  will 
appear  later  that  in  general  a  number  has  two  square  roots  not 
equal  to  each  other,  three  cube  roots  not  equal,  and  so  on. 

\^  The  theorems  that  follow  in  this  chapter,  though  uni- 
versally true,  are  here  proved  for  commensurable  numbers  only  : 
for  incommensurables  see  VII.  §  7,  and  for  imaginaries,  see 
X.  §§  2,  3. 


1.  §4.]     ADDITION  COMJMTJTATIVE  AND  ASSOCIATIVE.         35 


§  4.     ADDITION    COMMUTATIVE    AND    ASSOCIATIVE. 

Theor.  1.  The  sum  of  two  or  more  numbers  is  the  same,  in 
whatever  order  the  numbers  are  added,  and  however  they  are 
grouped. 

Let  ■•"«,■*■&,...  be  any  positive  integers,  ~m,  "w,  ...  any  nega- 
tive integers,    (- j,    (-),...  any  simple  fractions  ;  then  will: 


=  •  •  •  whatever  the  order  or  grouping  of  the  terms. 
For  •.*  +a=      l  +  l  +  l  +  .--a  times  counted  on  (forward), 

+  6=      i_j_i_j_i_j ^  times  counted  on,    and  so  on, 

and  *.•  ~m  =  —  1  —  1  —  1 m  times  counted  off  (backward), 

"w=  —  1  —  1  —  1 n  times  counted  off,    and  so  on, 

+/r\          111 
and  •.•       -—.  H 1 j j r  times  counted  on, 

\XJ  XXX 

{-)  = "•  s  times  counted  off,  and  so  on, 

\yj       y    y    y 

.*.  the  whole  collection  of  units  and  parts  of  units,  being 
the  same,  counts  the  same,  on  and  off,  whichever 
unit  or  group  of  units,  part  or  group  of  parts,  is 
counted  first,  whichever  second,  and  so  on  ; 
and  that,  whether  the  units,  or  parts  of  units,  be  of  the 
same  value,  or  of  different  values.  Q.  e.  d. 

SUM  OF  OPPOSITES. 

Cor.  1.     The  sum  of  the  opposites  of  two  or  more  numbers  is 
the  opposite  of  their  sum. 

For      let  "^A,  +B, . . .  be  any  positive  numbers,  integral  or  frac- 
tional ;  and  ~m,  . . .  any  negative  numbers  ; 

then    •.•  -a4-"bH \-+m-] \-+a-{-+b-] h"MH 

=  -a++a+-b4-^bH h'^M  +  'MH =  0,  [th. 


■A4--BH \--^M-] and    +a+'*"bH I-"mH 

are  opposites.  Q.  e.  d.    [I.  §  6  df. 


36  PBIMAKY  OPERATIONS.  [II.  th. 

THEORY   OF   SUBTRACTION. 

Cor.  2.  If  to  the  minuend  the  opposite  of  the  subtrahend  he 
added,  the  sum  is  the  remainder. 

For      let  M,  s,  and  r  =  minuend,  subtrahend,  and  remainder ; 

then    •••  M  =  s+R,  [I.  §7  df. 

.-.  M  +  (-s)  =  R  +  s4-(-s)  [ax.  2 

=  R  +  (s  —  s)  [th. 

==R-fO       =R.  Q.  E.  D. 

Cor.  3.  If  there  he  a  series  of  additions  and  suhtractions,  the 
final  result  is  the  same,  in  whatever  order  they  are  performed, 
and  however  the  elements  are  grouped;  hut,  whenever  any  group 
is  made  to  follow  the  sign  of  suhtraction,  the  sign  of  each  element 
of  the  group  is  reversed. 

§  5.     SIGN    OF    PEODUCT. 

Theor.  2.  If  the  multiplier  he  positive,  the  product  is  of  the 
same  sense  as  the  multiplicand;  if  negative,  of  the  opposite  sense. 

For    *.*  multiplication  b}'  a-{  ^  o-  t'  -    Diultiplier  is  a  repeated 

^  subtraction  ^^  *^^  multiplicand,  or  of  one  of  the 

equal  parts  of  it,  ■{  ^,       the  modulus  0,     [I.  §  8  df. 

and  •.•  subtraction  of  any  number  from  0  gives  the  opposite  of 
that  number  ;  [th.  1  cr.  3 

.-.the  sense  of  the  ^  remainder  (P^<^^^^^)  ^  is  "^^     changed 
thereb}'.  q.  e.  d. 

even 
Cor.  1.    If  the  numher  of  negative  factors  he-{     , , '  the  prod- 
uct is  ^  P^^^^''^^' 
»  negative. 

Note.    In  this  corollary  0  is  counted  an  even  number. 

Cor.  2.    In  division,  if  the  divisor  6e-{  ^^„,-  j  the  quotient  is 

of  the -l  ^^^^  .,    sense  i  f^  the  dividend. 
-'        '  opposite  '  to 

Note.  Th.  2  and  Cor.  2  are  summarized  in  the  familiar  rule 
for  sign  of  product  or  quotient :  "  Like  signs  give  -f ;  unlike,  — ." 


2,  3.    §  6.]     IVIULTIPLICATION  COlVOnXT.   AKD  ASSOC.  37 

§  6.    MULTIPLICATION  COMMUTATIVE  AND  ASSOCIATIVE. 

Theor.  3.  The  product  of  two  or  more  numbers  is  the  same, 
in  whatever  order  the  factors  are  multiplied,  and  however  they 
are  grouped, 

(a)   Two  positive  integers,  a,  b  ;  then  will  a  X  b  =  b  x  a. 

For      let     ♦         *         *        •••         * 
♦♦*•••♦ 

♦**•••* 

be  a  collection  of  stars,  trees,  or  any  other  units,  con- 
sisting of  a  horizontal  lines,  and  h  vertical  columns  ; 
then    *.*  if  a,  the  number  of  stars  in  one  column,  be  multiplied 
by  h,  the  number  of  columns,  the  product,  a  X  b, 
gives  the  whole  number  in  the  collection  ; 
and    *.*  if  6,  the  number  of  stars  in  one  line,  be  multiplied  by 
a,  the  number  of  lines,  the  product,  b  X  a,  gives 
the  whole  number  in  the  collection  ; 
r.  a  xb  =  b  X  a.  q.e.d.  [ax.  1 

(6)   TJiree  positive  integers,  a,  b,  c  ;  then  will 

axbxc=bxaxc=cxaxb=cxbxa 


■  axcxb  =  cxaxbz=bxaxc  =  bxcxa 


=  b  xcx  a  =  cxb  xa  =  ax  bxc  —  axcxb. 
For      let    a        a        a       ...       a 
a        a        a       ...        a 

a        a        a       ...       a 

be  a  collection  of  groups  of  a  units  each,  in  b  hori- 
zontal lines  and  c  vertical  columns  ; 
and     '.'  a  xb         is  the  number  of  units  in  one  column, 


.'.  axb  X  c  is  the  number  of  units  in  all  the  c  columns, 
i.e.,  in  the  whole  collection  ; 
then    '.-axe         is  the  number  of  units  in  one  line, 


ax  cxb  is  the  number  of  units  in  all  the  b  lines, 

i.e.,  in  the  whole  collection  ; 
ax6xc  =  axcx6.  q.e.d.    [ax.  1 


38  PKBIARY  OPERATIONS.  [11-  t^. 

So       let    6        6        h       ...       h 
b        b        b       ...       b 

b        b        b       ...       b 

be  a  collection  of  groups  of  b  units  each,  in  a  lines 
and  c  columns ; 
then   *.•  each  of  the  c  columns  has    &  x  a  units, 
and     *.*  each  of  the  a  lines  contains  6  x  c  units, 

.*.  6  X  a  X  c  =  b  xcxa.  q.e.  d. 

But*.*  a  X  6  =  6  X  a,     ax  c  =  cxa,     bxc  —  cxb^ 


and    axbxc  =  c  X  axb^    axcxb  =  b  xaxc, 


b  xcxa=axb  xci  [(a) 

axbxc=bxaxc=cxaxb=cxbxa 


=  axcxb  =  cxaxb=:ibxaxc  =  bxcxa 

=  b  X  cxa  =  cxb  xa=ax  b  x  c  =  ax  c  x  b.     q.e.d. 

Note.     In  this  proof  the  reader  wiU  observe  that  the  bar  can 

be  placed  over  a  group  of  factors,  or  removed,  at  pleasure,  when 

that  group  stands  at  the  left  end  of  the  series,  but  not  otherwise. 

E.g.,   axbx  c  =  a  X  6  X  c; 
for  in  either  case,  the  product  of  a  by  &  is  first  found  and 

then  that  product  is  multiplied  bj^  c. 
But      a  X  6  X  c  is  a  ver}^  different  matter  ; 
for.  in  this  case,  the  product  of  b  by  c  is  first  found,  and  a 

is  then  multipUed  by  this  product. 

(c)  Any  number  of  positive  integers. 

1.  The  theorem  is  true  for  three  factors.  [(a,  b) 

2.  If  it  be  true  up  to  n  factors  inclusive,  it  is  true  also  for 
n  + 1  factors. 

For  let  the  n 4- 1  factors,  a,  b,  c, ...  i,  J,  7j,  be  grouped  and 
multiplied  together  in  any  desired  way,  and  let  the 
product  be  p ; 
then  *.•  p  is  got  by  multiplying  the  product,  say  q,  of  some  of 
these  factors,  by  the  remaining  factor,  or  by  the 
product,  say  k,  of  the  remaining  factors, 
.'.  p  =  Q  X  R. 


3.  §6.]  MULTirLlCATION  CO^ENIUT.   AND   ASSOC.  39 

Let      R  be  that  one  of  these  products  which  has  the  factor  A;, 
and  let     s  be  the  product  of  the  other  factors  of  r  ; 
then   •••  neither  Q,  R,  s  nor  the  product  Q  X  s  has  more  than  n 
factors, 
.*.  in  each  of  them  the  several  factors  may  take  any  desired 
order,  [ti3T-  above 

.'.  p,  =  Q  X  R,  =  Q  X  s  X  A; 

=  Q  X  s  X  k  [(6) 

=  «  X  6  X  •  •  •  X  j  X  k 
=  axbX'"Xj  Xlc. 

3.   But  the  theorem  is  true  for  three  factors,  [(a,  6) 

.-.  it  is  true  for  four  factors  ;  [2,  above 

So        for  five  factors,  for  six  factors,  ....  q.  e.  d. 

Note.     This  proof  is  an  example  of  proof  by  induction.     [§  1 

(d)  Any  number  of  integers,  '•"a,  "b,  ''"c,  . . .,  "'"1,  "m,  ~n,  whereof 
k  factors  are  negative. 

For      in  whatever  order  the  factors  are  multiplied, 

p,  their  product,  =  ±'*'a  X  "'"6  X  •  •  •  X  """m  x  """w,         [(c) 
_,  positive  ^        ^      ^.  .^  even.  Q  e  d. 

'  negative  '  odd.     [th.  2  cr.  1 

(e)  Any  number  of  factors,  a,  b,  c,  ...-,-, ...,  whereof  some, 

XV 

or  all,  are  simple  fractions.  -^ 

For-.-  in  the  product  (XX&XCX---X^X-X  •••, 

2/      y 

=  ±+ax+5  X+cX  •••  X     -    X     -    X---, 

\^/     \yJ 

each  unit  of  the  product  a  x  6  X  c  x  •  •  •  is  divided 

into  X  equal  parts  and  r  of  them  are  taken, 
and  then  each  of  these  axb  x  ex  -•'  Xr  parts  is  divided  into 

y  equal  parts,  and  s  of  them  are  taken,  and  so  on, 
i.e.,  the  unit  is  divided  into  xxyx  •••  equal  parts,  and 

ax&XcX---Xrxsx---of  them  are  taken  ; 

^   ,    ,  r      s  axbxcx  •••  X  rxs--- 

.'.  axbxcx--'X-X-X"'  = 


X     y  xxyx 

s  r 

K  -Xb  X-XCX'-'  each  of  t 

y  ^ 

is  divided  into  y  parts,  and  s  of  them  are  taken, 


s  r 

So        in  the  product  ax -Xb  x-Xcx---  each  of  the  a  units 

y  X 


40  PEIMARY  OPEKATIONS.  [II.  th. 

and  then  each  of  these  ax  s  parts  is  taken  b  times, 

and  then  each  of  these  ax  sx  b  parts  is  divided  into  x  equal 

parts,  and  r  of  them  are  taken, 
and  then  each  of  these  ax  s  xb  xr  parts  is  taken  c  times,  and 

so  on; 
i.e.,  the  unit  is  divided  into  yxxx---  equal  parts, 

and  axsxbxrxcx  •••of  them  are  taken. 


y  X  yxxX'" 

But-.-  2/Xa;x  •••  =  XX2/X  •••,  [(d) 

and  axsxbxrxcx '"  =  axbxcX"'Xrxs', 

.*.  in  each  of  these  two  products,  the  unit  is  divided  into 
the  same  number  of  equal  parts,  and  the  same 
number  of  these  parts  are  taken. 
So        for  all  other  possible  products  of  these  factors  ; 

.-.  the  products  are  all  equal.  q.e.d. 

(/)   Any  number  of  factors^  whereof  some  or  all  are  neither 
integers  nor  simple  fractions,  but  which  are  all  commensurable. 

For,  let  A,  B,  c, ,  -, ...  be  the  values  of  such  factors  when 

X      Y 

reduced,  wherein  the  letters  all  stand  for  integers  ;         [I.  §  1  df. 
then    -.-  the  value  of  the  product  axbXCX--^X  —  X-X  ••• 

X        Y 

is  the  same,  whatever  the  order  or  grouping  of  the 

factors,  [(d,  e) 

.-.  the  product  of  the  given  factors  is  the  same,  etc.  q.  e.  d. 

THEORY  OF  FRACTIONS. 

Cor.  1 .    TJie  prodtLct  of  two  or  more  fractions  is  a  fraction 
J        ,  numerator      .    .,  j    *    ^  *?    »  numerators        ^  ,, 

""^"'^  -!  denominator  ''  *^'  P'"^"^  "f  *^  ^  denominators  "^  *^ 

given  fractions. 

n   n    n 
Let       -?  —:?  — r,  . . .  be  any  fractions, 
d   d'   d" 

.,  •n  ^  V,  ^'  V,  n  X  n'  X  ••• 

then  will-  X  -  X      •  =  - — -. 

d      d'  dxd'  X  •" 

(a)  n,  d,  ...  all  integers. 

This  case  was  proved  in  the  demonstration  of  Th.  3  (e) . 


3.  §6.]  MULTIPLICATION  COMMUT.  AND  ASSOC.  41 

(b)  The  fractions  and  their  numerators  and  denominators  any 
commensurahles  whatever, 

n    n 
For,  let  /,  /',  ...  be  the  values  of  -,  — ,  ...  respectivel}' ; 

a   a 

then    '.'  n=fxd,     n'=/' x  c?',  ...  ,  [I.  §  9  df. 

.-.  nxn' X"'={fXd)x{f'xd')X"'  [ax.  4 

=  (/X/'X---)x((^  XcZ'x---),         [th.3 

"  dxd'X"-~~^''\'' 

=  '^'x^X*---  Q-^-^-     [I-§9df. 

d      d' 

Cor.  2.    The  reciprocal  of  any  fraction  is  the  same  fraction  in- 

.   ,     .  ^     ,•         T        I  numerator      .    ,,     ,  denominator 

verted;  i.e.,  a  fraction  whose  {   ,  .     .     is  the< 

■'        '     -^  '  denominator  '  7iumerator 

of  the  given  fraction. 

For,  let  n  and  d  be  any  commensurable  numbers ; 

then   •.•  ^x^  =  ^-^  =  l, 
71      d     nxd 

d  n 

.'.  -  is  the  reciprocal  of  —  q.  e.d.     fl.  §8  df. 

n  d 

CoR.  3.  The  quotient  of  one  number  by  another  is  the  product 
of  the  first  by  the  reciprocal  of  the  other. 

(a)   The  divisor  and  dividend  both  simple  fractions. 
For,  let  -  and  —  be  any  two  fractions, 

then  •  .  /'^  y  A  y  ^^_^  y  f(^'  y  '^'\  _ '^  y  d'xn'_n. 
*^'^    •   [d''n')''d-d''[n'''d')-d''^;j^-r 

n    n'      n      d'  ft   c  a  i^. 

•••  rw=a''^-  ^•'=-''-  ^^■^'^'- 

Note.  Cors.  1,  2,  3  lead  to  the  reduction  of  complex  prod- 
ucts and  quotients  of  commensurahles  to  simple  fractions,  either 
directly  or  by  progressive  steps.  By  Cor.  2,  the  reciprocal  of 
any  commensurable  is  a  commensurable ;  hence,  by  Cor.  3,  if 
both  numerator  and  denominator  be  commensurahles,  so  is  the 
fraction  ;  by  Cors.  1,3,  any  product  or  quotient  of  such  frac- 
tions is  got  as  a  simple  fraction,  and  so  on.  Compare  Th.  5, 
Cor.  8. 


42  PRniARY  OPERATIONS.  [II.  th. 

(6)    Tlie  dividend  and  divisor  any  commensurahles  whatever. 
For,  let  N,  D  be  any  two  eommensurables  ; 
then  •.*    D  is  commensurable, 

.'.    -is  likewise  commensurable;  [(a)  nt. 

.-.  (n  X  - ]  X  D  =  N  X  [-  X  D  J  =  N  ;  '\th.  3 

.-.    Nxi  =  N:D.  Q.E.  D.    [I.  §9clf. 

D 

Cor.  4.    The  product  of  the  reciprocals  of  two  or  more  numbers 

is  the  reciprocal  of  their  product. 

r    9 
For,  let  a,  6,  ...-,-,...  be  any  numbers, 
X   y 


r      s 

x-x-x 

X      y 
-X-)X  ••• 


then  •.•  (_x-rX  •••X-X- X  •••  )  X  f  a  X  &  X  ••< 
\a     0  r     s  J      \ 

=  I  -  X  a  )  X  f  r  X  6  )  X  •  •  •  X  ( -  X  -  J  X 

\a       J      \b       J  \r     x)      \s      y 

=  1; 

11  X     'u  ,  ,  r      s 

.-.    -x-X-'-X-X-X---  and  ax6x  •••  X  -X-X  ••• 
a     h  r      s  ^     y 

are  reciprocals.  q.  e.  d. 

Cor.  5.  If  the  numerator  of  a  fraction  he  multiplied  by  any 
number^  the  fraction  is  multiplied  by  that  number. 

For,  let  -  be  any  fraction,  and  k  any  multiplier ; 
a 

then         ^?L^=(nxA;)x-  =  fnxiV^'  =  3X^-    ^'^''^' 
d  '     d      \       dj  d 

Note.  In  this  corollary  and  the  two  corollai:ies  that  follow, 
«'  multiplied  by"  includes  "  divided  by,"  since  to  divide  by  k  is 
but  to  multipl}'  by  its  reciprocal. 

CoR.  6.  If  the  denominator  of  a  fraction  be  multiplied  by  any 
number^  the  fraction  is  divided  by  that  number. 

For,  let  -  be  any  fraction,  and  k  any  multiplier ; 

then         _!L.       „x-J-  =  »x('ixi)  =  f«xl')xi  =  ^:fc. 
dxk  dxk  \d     kj      \       dj     k     d 


3.  §  6.]  MULTIPLICATION  COINOIUT.  AND  ASSOC.  43 

Cor.  7.  If  both  terms  of  a  fraction  he  multiplied  by  the  same 
number,  the  value  of  the  fraction  is  not  changed. 

n 
For,  let  -  be  any  fraction,  and  k  any  multiplier ; 
a 
,,  n  xk      n      k      n  p^ 

then         -^l-  =  -x-  =  -.  Q.E.D.     [or.  1 

dxk     d     k     d  ■- 

Cor.  8.  If  there  be  a  series  of  multiplications  and  divisions, 
the  final  result  is  the  same,  in  whatever  order  they  are  performed, 
and  however  the  elements  are  grouped;  but  whenever  any  group 
is  made  to  follow  the  sign  of  division,  the  sign  of  operation  of 
each  element  of  the  group  is  reversed. 

Note.  The  reader  will  observe  the  analogy  between  this  cor- 
ollary and  Th.  1,  Cor.  3.  He  will  see  that,  if  three  or  more 
numbers  are  joined  b}^  the  signs  X  and  ;,  he  may  introduce  or 
remove  brackets  just  as  if  they  were  joined  by  the  signs  -f  and  — . 

QUOTIENT   OF  A  PRODUCT  BY  ITS  FACTORS. 

CoR.  9.  If  the  product  of  several  factors  be  divided  by  one  of 
them,  or  by  the  product  of  two  or  more  of  them,  the  quotient  is  the 
product  of  the  rernaining  factors. 

For    *.•  the  product  of  the  remaining  factors  by  the  divisor  is 

the  product  of  all  the  factors,  [th.  3 

.*.  the  product  of  the  remaining  factors  is  the  quotient  of 

the  product  of  all  the  factors  b}^  the  divisor,  q.  e.  d. 

PRODUCT  OF  INTEGRAL  POWERS. 

Cor.  10.  TJie  product  of  two  or  more  integral  powers  of  any 
same  number  is  a  power  of  that  number  whose  exponent  is  the 
sum  of  the  exponents  in  the  factors. 

For,  let  A  be  any  number,  and  I,  m,n  ...  any  positive  integers, 
then   *.•  A*  =  lXAXAXAX---?    times, 
A'"=lXAXAXAX  '"  m  times, 
and         A-"=l   :  A  :  A  :  A  :  •••n  times,  and  so  on.  [I.  §10df. 
.•*  a'  X  A*"  X  A-"  X  ••• 

=  (1  X  A  X  A  X  •  •  •  ?  times)  x(lXAXAX---m  times) 
X  (1  :  A  :  A  :  •  •  •  w  times)  X  •  •  • 

=  lXAXAX---(Z  +  mH )  times 

:  a:  a:  •••{n-\ )  times 

=  1  XAXAX"-(Z-|-mTJ n )  times  [cr.  9 

s=  A*  ■'■'""'*"•.  Q.E.D. 


44  PRIMABY  OPERATIONS.  [II.  th. 

Cor.  11.  An  integral  power  of  any  integral  power  of  a  base^ 
is  that  power  of  the  base  tuhose  exponent  is  the  product  of  the 
two  given  exponents. 

Let  Abe  an}'  number,  m  and  n  any  integers,  then  will  (a")'"= a*"**. 

(a)  m  positive. 
For         (a*)"^'",  =  1  X  a" X  a- X  •••  m  times,  [I.  §  10  df. 

_^n  +  n+...m  timet  [CF.  10 

=  A*"".  Q.  E.  D. 

(6)  m  negative. 
For         (a**)""*,  =  1 :  a"  :  a"  :  ...  +m  times,  [I.  §  10  df. 

_  1  .  ^n  +  n+...+m time.  [cr.  10 

=  1  :  a"^*"* 

=  a""*",  or  =  A"***  if  the  sign  of  quality  be  erased. 

,  Q-E.D. 

Cob.  12.  The  {  ^^^  "^  .  of  like  integral  powers  of  two  or  more 
numbers  is  the  same  power  of  the  \  ^I^^^r^*  of  those  numbers. 

Let  A,  B,  c,  ...  be  any  numbers,  n  any  integer,  positive  or 
negative,  then  will  a"  X  b"  :  c"  •  •  •  =  a  X  b  :  c  •••   . 

(a)  n  positive. 
For   *.•    A"=lXAXAX...n  times, 
B"=lXBXBX.--n  times, 
c*=lxcxcx-.-n  times,     and  so  on ; 
.*.    A*  X  B"  :  C**  ••• 

=  (lXAXAX...w  times)  x(lXBXBX-..n  times) 

:  (1  X  c  X  c  X  •  •  •  w  times)  •  •  • 
=  1  X  AX  B  :  c  •«.  X  AX  B  :  c  •••  X  •  •  •  w  times 


=  A  X  B  :  c  •••    .  Q.E.D. 

(6)  n  negative. 
For    •.*  a""  =1  :  A  :  A  :  •  •  •  +n  times, 
b"**  =  1  :  b  :  b  :  • .  •  ■*'n  times, 
c~"  =  1  :  c  :  0  :  .  •  •  +71  times,     and  so  on ; 
.*.  A~"X  B~"  :  c""-" 

=  (1 :  A  :  A :  •  •  •  +71  times)x  (1 :  b  :  b  :  • . •  +7i  times) 

:  (1 :  c  :  C  :  •  •  •  "^n  times)  •  •  • 
=  1  :  AX  B  :  0  .••  :  A  X  b  :  c  •••  :  •  •  •+w times  [cr. 8 
=  (aXB  :  C  •«.)"'".  /%  X.  ^ 


4.  §  7.]  MULTIPLICATION  DISTRIBUTIVE  AS  TO  ADDITION.  46 


§  7.     MULTIPLICATION  DISTEIBUTIVE  AS  TO  ADDITION. 

Theor.  4.    The  sum  of  two  or  more  like  numbers  is  the  product 
of  the  common  factor  by  the  sum  of  the  coefficients, 

~r 
Let  "*"m  -a,    ^w •  a,    "p  •  a,    —  a,  ...  be  any  like  numbers, 

^  ± 

whereof  a  is  the  common  factor,  and  "''m,  +n,  ~p,  — ,  ...  are  the 

coefficients ; 

-r 
then  will  "♦"m«a++n-a  +  "p-a  +  -7--aH 

=  (+m  ++71  +"i)  +  —  H )  •  a. 

For  *.•  -^m'a=     a-{-a-\-a-\ counted  on  +m  times, 

+n-a=      a  +  a  +  aH counted  on  +?i  times, 

-p,a  =  —  a  —  a  —  a counted  off  '+p  times, 

—  a  =  the  icth  part  of  a    counted,  on  or  off,  r  times, 


.  .  "*'m •  a -h+n •  a  -^~p 'a-] a  -\-  -•- 

=  a-\-a-\-a-\ counted  (+m  ++W  -f"jp  •  •  •)  times 

±  the  -th  part  of  a  H ; 

.•.     the  whole  sum  is("*"m+''"n+"p+— -H )  -  a.  q.e.d. 

Cor.  The  sum  of  two  or  more  fractions  having  a  common 
denominator  is  a  fraxition  whose  numerator  is  the  sum  of  their 
numerators^  and  whose  denominator  is  the  common  denominator. 

For,  let-,  ±— ,  ...  be  any  fractions  having  a  common  de- 


nominator. 

then          l±%  +  - 
d      d 

■K"5)*(-«i)- 

[th.  3  cr.  3 

=  (n±n'+'")xl 

Oj 

[th. 

_n±n'H 

Q.E.D. 

d 
Note.    In  this  corollary,  and  in  general,  subtraction  is  but  a 
case  of  addition. 

Ti'  n 

E.g.,  to  subtract  the  fraction  —  is  to  add  its  opposite,  —  —■• 

d  d 


46 


PREVIAEY  OPERATIONS. 


[n.  th. 


Theor.  5.  The  product  of  two  or  more  polynomials  is  the  sum 
of  the  several  products  of  each  term  of  the  first  factor  by  each 
term  of  the  second  factor  by  each  term  of  the  third  factor,  and 
so  on. 

(a)  Two  factors,  a  +  b  H f-  -  +'-'and  a'+b'H f--  +  •  •  • , 

X  x' 

wherein  a,  b,  ... ,  a',  b',  ...  are  any  integers, positive  or  negative, 


and  -1 


r    r' 


X     X' 


are  any  simple  fractions. 


For       (a  +  &  +  ---  +  -  +  ---)  X  (a'  +  6'  +  ...+^  +  ---) 
X  a;' 

=  (a-|-6H 1 1 )  counted  "^a'  times,  on  or  off, 


4-(a  +  5H 1 1 )  counted  """fe'  times,  on  or  off, 

X 

+ 

+  the  -^th  pai'tof  (a-\-b-\ !--  +  •••)»  on  or  off, 

a;'  X 

+ 

=  a  counted  a'  times  +  b  counted  a'  times  H + 

counted  a'  times  -\ 

+  a  counted  6'  times  +  b  counted  6'  times  -\ + 

counted  6'  times  H 


+  the  -th    part    of    a  +  the    l-th  part  of   6  + 
aj'  a' 

+  the  -,th  part  of  -  +  ..- 
a;'  X 


=  axa^  +  6xa'  + 

+  ax6'+6x6'  + 

+ 

r'               r' 
x'              x' 
+ 


+  T^xa^  + 


+  -X^^   + 


,    r      r     , 

+  -  X  -   + 

X      x' 


Q.E.J). 


Note.  Manifestl}',  if  a  term  in  either  factor  is  negative,  the 
corresponding  partial  product  is  negative  or  positive  according 
as  the  co-factor  of  this  term  is  positive  or  negative. 


6.  §7.]        MULTIPLICATION  DISTRIB.  AS  TO  ADDITION.         47 

(6)    Three  or  more  factors. 
For     *.•  the  product  of  two  factors  is  the  sum  of  the  partial 
products  of  each  term  of  one  factor  by  each  term 
of  the  other,  [  (a) 

and  '.*  the  product  of  this  product  by  a  third  factor  is  the  sum 
of  the  partial  products  of  each  term  of  this  product 
b}^  each  terra  of  the  third  factor ;  [  (a) 

.*.  the  product  of  three  factors  is,  etc.  q.e.d. 

So,       for  any  number  of  factors.  q.  e.  d. 

FORM   OF  PRODUCT. 

CoR.  1.  The  form  of  a  product  is  independent  of  the  values 
of  the  letters  that  enter  into  it;  i.e.,  the  same  numerals,  letters, 
exponents,  coefficients,  and  signs,  occur  and  combine  in  the  same 
order,  whatever  the  numbers  for  which  the  letters  stand. 

Cor.  2.  If  each  factor  be  symmetric  as  to  two  or  more  letters. 
the  product  is  also  symmetric  as  to  the  same  letters. 

CoR.  3.  If  any  values  be  given  to  the  letters,  or  if  any  definite 
relations  be  assumed  between  their  values,  the  value  of  the  prod- 
uct equals  the  product  of  the  values  of  the  factors. 

CoR.  4.  The  sum  of  the  coefficients  of  a  product  is  the  con- 
tinued product  of  the  sum  of  the  coefficients'  of  the  first  factor,  by 
the  sum  of  the  coefficients  of  the  second  factor,  and  so  on. 

CoR.  5.  The  degree  of  the-{  j  ,f  term  of  a  product,  as  to 
any  letter  or  letters,  is  the  sum  of  the  degrees  of  the  {  ■.  ^     .   terms 

of  the  factors,  as  to  the  same  letter  or  letters.     In  particular,  the 
degree  of  the  product  is  the  sum  of  the  degrees  of  the  several 
.  factors. 

CoR.  6.  If  each  factor  be  homogeneous  as  to  any  letter  or  let- 
ters, then  the  product  is  homogeneous  as  to  the  same  letter  or  letters. 
CoR.  7.  The  ivhole  number  of  terms  in  any  product,  before 
reduction,  is  the  continued  product  of  the  number  of  terms  in  the 
several  factors ;  and  the  product  of  two  or  more  polynomials  can 
never  be  reduced  to  iess  than  two  terms;  viz. :  the  term  of  highest 
degree  and  the  term  of  lowest  degree  as  to  any  letter  or  letters. 


48  PKIMARY  OPERATIONS.  [II.  th. 

Cor.  8.    The  value  of  evenj  rational  expression  whose  elements 
are  commensurable  numbers  is  a  commensurable  number. 
For      •.•in  such  an  expression  the  elements  enter  only  as  ele- 
ments of  sums,  ditferences,  products,  quotients, 
and  integral  powers, 
and       •.*  these  results  enter  only  as  elements  of  new  sums,  etc., 

and  so  on, 
and      *.'  the  sums,  etc.,  of  commensurables  are  commensur- 
ables ;  [th.  3  cr .  3  nt. ,  th.  4  cr. 

.'.  the  sums,  etc.,  of  the  elements  are  commensurables, 
.*.  the  sums,  etc.,  of  these  results  and  the  original  elements 

are  commensurables,    and  so  on  ; 
.*.  the  final  result  is  commensurable.  q.  e.  d. 

§  8.    PROPORTION. 

Four  numbers  are  propoHional  (in  proportion)  when  the  first 
is  such  multiple,  part,  or  parts,  of  the  second,  as  the  third  is 
of  the  fourth  ;  i.e.,  when  the  quotient  of  the  first  by  the  second 
equals  the  quotient  of  the  third  by  the  fourth. 

E.g.^  if  a  :  6  =  c :  d,  then  a,  6,  c,  d  are  proportionals,  taken  in 
the  order  given. 

Q'        C 

A  proportion  is  also  written  in  the  forms  a:b::  c:  d  and  t  =  -;i 

and  it  is  read :  a  is  to  h  as  c  is  to  d,  or  the  ratio  of  a.  to  h  equals 
the  ratio  ofctod,  or,  more  briefly,  a  ^o  b  equals  c  to  d. 

These  quotients  are  now  called  ratios;  the  dividends,  ante- 
cedents; the  divisors,  consequents;  the  first  and  fourth  tenns, 
extremes;  the  second  and  third  tenns,  means;  the  fourth  term, 
a.  fourth  proportional  to  the  other  three. 

Three  numbers  are  proportional  when  the  quotient  of  the  first 
by  the  second  equals  the  quotient  of  the  second  by  the  third.  It  is 
a  case  of  four  proportionals  wherein  the  two  means  are  the  same 
number.  The  second  number  is  a  mean  proportional  between 
the  first  and  third,  and  the  third  is  a  third  proportional  to  the 
first  and  second. 

E.g.,  a:b  =  b:c,  wherein  6  is  a  mean  proportional  between  a 
and  c,  and  c  is  a  third  proportional  to  a  and  6. 


6.  §  8.]  PROPORTION.  49 

Six  or  more  numbers  are  in  continued  proportion  when  the  first 
is  to  the  second  as  the  third  is  to  the  foiuth,  as  the  fifth  is  to  the 
sixth,  and  so  on.     E.g.^  a:h  —  c:d  =  e  :/=  •••. 

By  aid  of  Th.  7  (6)  this  proportion  ma}^  be  written  in  the  form 
a  :  c  :  e  :  •••  =  6  :  d  :/:  •••,  wherein  a,  c,  e, ...  are  the  antecedents, 
and  6,  d,  /,  ...  the  consequents.  This  notation  must  not  be  con- 
founded with  that  used  on  p.  43  and  elsewhere. 

Theor.  6.  If  four  numbers  he  proportional^  the  product  of  the 
extremes  equals  the  product  of  the  means;  and,  conversely,  if  the 
product  of  two  numbers  equal  the  product  of  two  others,  the  four 
numbers  form  a  proportion,  ivherein  the  factors  of  one  product 
are  the  extremes  and  the  factors  of  the  other  product  are  the  means. 

(a)  Let  a  :  b  =  c :  d,  then  will  ad  =  be. 
For     •. •   (a:b)xbd  =  {c:d)x  bd,  [ax.  4 

.-.  ad  =  bc.  Q.  E.  D.    [th.  3cr.  9 

(6)  Let  ad  =  be,  then  will  a:b  =  c:d. 
For     •.•  ad:bd  —  bc:  bd,  [ax.  5 

.*.  a:b  =  c:d.  Q.  e.  d.    [th.  3  cr.  7,  cr.  5  nt. 

y~«  extrcTne 

Cor.  1.    If  four  numbers  be  proportional,  either  ■{  is 

the  quotient  of  the  product  of  the  ^  *^^,^^^     by  the  other  -l  ^^^^'^^' 
^  ^        ^  -f        \  extremes  ^  »  mean. 

For,  let  a  :  6  =  c  :  d, 

then   •.•  ad  =  bc,  [th. 

.-.  a   =bc:d,    b  =  ad:c,    c  =  ad:b,    d  =  bc:a.     [ax.  5 

CoR.  2.   If  three  numbers  be  proportional,  either  extreme  equals 

the  quotient  of  the  square  of  the  mean  by  the  other  extreme,  and 

the  mean  equals  the  square  root  of  the  product  of  the  extremes. 

For,  let  a  :  b  =  b  :  c, 
then   *.•  ac  —  W,  [th. 

.-.  a  =b^:c,    b=-y/ac,    c  =  b^:a.         Q.  e.  d.    [ax.  5,7 

Note.  The  equation  ad  =  be  may  be  resolved  into  eight  dif- 
ferent proportions,  four  of  them  with  a  and  d  for  extremes  and 
b  and  c  for  means,  and  four  of  them  with  b  and  c  for  extremes 
and  a  and  d  for  means.  The  reader  may  write  them  out ;  he  will 
find  two  of  them  given  in  Th.  7  (a,  b) . 


50  PBIMABY  OPERATIONS.  [II.  th. 

Theob.  7.    If  four  numbers  he  proportional^  they  are  propor- 
tional : 

(a)  Inversely:  the  second  to  the  first  as  the  fourth  to  the  third. 
Let  a:h  =  c:d^  then  will  b:a  =  d:c. 

For     •.'  ad    =bc,  [th.  6 

.-.  b:a  =  d:c.  Q.  e.d.     [th.  6  cv. 

(b)  Alternately:  the  first  to  the  third  as  the  second  to  the 
fourth. 

Let  a'.b  =  c:d^  then  will  a:c=b:d. 
For    '.-ad    =bc,  [th.  6 

.*.  a:c  =  b:d.  q.e.d.    [th.  6  cv. 

(c)  By  addition  or  subtraction  (composition  or  division)  : 
the'{  ^^  .    ,     of  the  first  ^  ^'^^  the  second,  to  the  first  or  second, 

^  ^^^  <  remainder  ""^  *^^  ^^''^  ^  Zt  ^^'^  •^^^'*^^'  ^^  ^^'^  ^^''"^^  ""' 
fourth. 

Let  a :  b  =  c :  d,  then  will 

a±b:a  =  c±d:c,   and  a±b  :b  =  c±d:  d. 
For     •.•  ad=bc,  [th.  6 

.-.  ac  ±bc  =:ac±ad     and   ad±bd  =  bc±  bd,    [ax.  2,  3 
i.e.,        (a±b)c  =a{c±d)   and  {a  ±b)d  =b(c±  d), 

.*.  a±b:a  =  c±d:c    and   a±b:b=c±d:d.    q.e.d. 

(d)  ^2/  oddition  and  subtraction  (composition  and  division) : 
the  sum  of  the  first  and  second  to  their  remainder  as  the  sum  of 
the  third  and  fourth  to  their  remainder. 

Let  a:b  =  c:d,  then  will  a-\-b:a  —  b  =  c-\-d:c  —  d. 
For     '.'  a-\-b:a  =  c-{-d:c    and    a  — 6:a  =  c  —  d:c,        [(c) 
.'.  a-^b:c-{-d  =  a:c    and    a  —  b:c-'d  =  a:c,        [(6) 
.-.  a-{-b:c-\-d  =  a  —  b:c  —  d,  [ax.  1 

.'.  a  +  b:a—b  =  c-\-d:c  —  d.  q.e.d.    [(6) 

CoR.  Conversely,  if  four  numbers  be  proportional,  (a)  in- 
versely, (b)  alternately,  (c)  62/  addition  or  subtraction,  or  (d)  6?/ 
addition  and  subtraction;  then  is  the  first  to  the  second  as  the 
third  to  the  fourth. 

The  reader  may  prove,  hj  retracing  the  steps,  from  conclusion 
to  data,  in  each  of  the  above  demonstrations. 


7-9.  §8.]  PROPOETIOISr.  51 

Theor.  8.   If  there  he  two  or  more  sets  of  proportionals,  the 
products  of  their  corresponding  terms  are  proportional. 
Leta:6  =  c:d,    a' :  6' =  c' :  d',    a"  :  6"=  c"  :  d",  ••., 
then  will  aa^a^^  •  •  •  :  66'6"  •  •  •  =  cdd'  •  •  •  :  dd'd^'  •  •  -. 
For     •.•  ad==^hc,   a'd'  =  b'&,    a"d"  =  b"c",--',  [th.  6 

.-.  ad'a'd''a"d"----  =  be •  b'c' - b"c" -- -,  [ax.  4 

.-.  aa'a"---'dd'd"---  =bb'b'--"  cc'c"---,  [th.3 

.-.  aa'a"--':bb'b"'--  =  cc'c"--- :  dd'd".--.  q.e.d.  [th.  6  cv. 
Cor.  1.   If  there  be  two  sets  of  proportionals,  the  quotients  of 
their  corresponding  terms  are  proportional. 
For,  let  a :  6  =  c  :  d    and  a' :  b'  =  c' :  d', 
then    *.•  ad       =bc      and  a'd'     =b'c',  [th.  6 

.     ad  be      .  ^     a    d       be  r       k  4.-U  o       i 

■■M'     =W'    *•'•'«'•  d= -ft'- ^'         ^^-S.tb-Scr.l 


Q.E.D.       [th.  6  CV. 


a''  b'~c'''  d'' 

CoR.  2.  If  four  numbers  be  proportional,  their  like  integral 
powers  are  proportional. 

The  reader  may  write  in  formula,  and  prove. 

Theor.  9.  If  six  or  more  numbers  be  in  continued  proportion, 
the  sum  of  the  antecedents  is  to  the  sum  of  the  consequents  as  any 
antecedent  is  to  its  consequent. 

Let       a:b  =  c:  d  =  e:f=  '",   then  will 

aH-c-|-e  +  ...  :6  +  d+/+.-.=a;6  =  c:d5=.... 
For    •.'  ad  =  bc,     af=be,     ••.,  [th.6 

.*.  ab-{-ad-\-af-\ =  ba -{- be -\- be -\ ,  [ax.  2 

...  a(6  +  d-f/+...>=Z>(a  +  c  +  e+...),  [th.4 

.*.  a  +  c  +  eH :b+d-\-f-\-"'=a:b.    q.e.d.    [th.6cv. 

CoR.  1.     i/"  a  :  b  =  c :  d  =  e  ;  f  =  •.., 

then         ha  +  ko  +  leH :hb  +  kd  +  lfH =  a:b, 

wherein   h,  k,  1,  •••  are  any  numbers. 

The  reader  may  state  in  words,  and  prove. 

CoR.  2.     i/"a:b  =  c:d  =  e:f=..., 
then         ha°  +  kc*^  -f  le°  +  ... :  hb"  +  kd°  +  1P+  ...  =  a° :  b°, 
vjherein   h,  k,  1,  •••  are  any  numbers  and  n  any  integer. 

The  reader  may  state  in  words,  and  prove. 


62  PEIMAEY  OPERATIONS.  [II.  pr. 

§  9.    PROCESS   OF   ADDITION. 
PrOB.  1.     To  ADD   TWO   OR  MORE  NUMBERS. 

(a)   The  numbers  like : 

To  the  common  factor  prefix  the  sum  of  the  coejfficieyits.    [th.  4 

E.g.,    10  ft.  down  +  20  ft.  up      +  CO  ft.  up      =  70  ft.  up, 

10  ft.  up      +  20  ft.  down  +  60  ft.  down  =  70  ft.  down. 
So,       10a;-15a;+20a;-25a;-h30a;=60a;-40a;=20a;,    [th.  2 

10ay-^20by  —  B0cy  =  {10a  +  20b  —  30c)y. 

(6)    The  numbers  unlike : 

Write  tJie  numbers  together,  vMh  their  proper  signs,  in  any 
convenient  order.  [th.  1 

E.g.,    19 xyz  —  29 mn  4-  39 a  —  49  is  irreducible. 
So,       10a2/-f-206?/  — 30c2/  is  usually  not  reduced,  but  may 
be  written  (lOa-f  206  -  30c)2/. 

(c)  Some  numbers  like  and  some  unlike : 
Unite  into  one  sum  each  set  of  like  numbers,  and  write  these 
partial  sums,  together  with  the  remaining  terms,  in  any  order. 

E.g.,    (a3+3a25  +  3a62-|.68)  +  (a3-3a2c  +  3ac2-c3) 
=  2a3  +  3a2(6-c)  +  3a(&2  4-02)4- (63  _c3). 

So,       3xy-\-7xy+^l^xy^lox^+^x^+y+^a^-^f 
n  0  0  0  0 

^o-1564-2c^  J   llm4-10^^y  ^  ^~^^/. 


§  10.    PROCESS   OF   SUBTRACTION. 

PrOB.  2.     To   SUBTRACT   ONE   NUMBER  FROM  ANOTHER. 

To  the  minuend  add  the  opposite  of  the  subtrahend,  [th.  X  cr.  2 

E.g.,   90 ft. up -60 ft. up  =  90 ft. up 4- 60 ft. down  =  30 ft. up, 

60ft.up-90ft.up  =  60ft.  up+90  ft.  down=  30ft.  down, 
90ft.  up-60ft.  down  =  90ft.  up+60  ft.  up  =  150ft.  up; 

i.e.,    +90 -+60  =  +30,    +60  - +90  =  "30,    +90--60  = +150. 

So,     [2a«  4-  3a2  (6  -  c)  +  3a  (62  4-  c2)  4-  (6^  -  c^)] 

-[a«-3a2c4-3ac2-(^]  =  a34-3a26  4-3a624-6^ 


1-3.  §  11.] 


PROCESS   OF  MULTIPLICATION. 


53 


Note  1 .    The  opposite  of  the  subtrahend  need  not  be  written ; 
but  the  sign  may  be  changed  and  the  addition  made,  mentally. 

Note  2.   The  definition  of  subtraction  leads  to  a  more  direct 
operation  : 

E.g.,    +8-  +3=  +5,  •.•  +8>  +3  by   +5, 

+8>   -3  by +11, 

-8<   +3by+ll,i.e.,-8>  +3by-ll, 

-8<   -3  by   +5,t.e.,-8>   "3  by  "5. 

+8<+10by   +2,i.e.,+8>+10by  "2, 

+8  > -10  by +18, 

-8<+10by+18,*.e.,-8>+10by-18, 

-8  > -10  by   +2. 


+8- 

+3  = 

■^5, 

+8- 

-3  = 

ni, 

-8- 

+3  = 

-11, 

'8- 

-3  = 

-5, 

+8- 

+10= 

-2, 

+8- 

-10  = 

+18, 

-8- 

+10  = 

-18, 

-8- 

-10  = 

-^2, 

§  11.     PROCESS    OF   MULTIPLICATION. 
PrOB.  3.     To   MULTIPLY   ONE  NUMBER  BY  ANOTHER. 

(a)  A  monomial  by  a  monomial : 

To  the  product  of  the  numerical  coefficients  annex  the  several 
literal  factors^  each  taken  as  many  times  as  it  is  found  in  both 
multiplicand  and  multiplier  together.  [th.  3 

Mark  the  product  ^  '^if  the  factors  are  taken  in  {  ^  ^^l^l^^y 

sense.  [th.  2  or.  1 

E.g., 


+9a6-3     x+7aV     =+63a36-V, 
-5  £c?/^  z~^  X'^1 0?  yz'"^  =~SDx'^y^z~'^, 
+da-'b^   X-7 
-5xy^z~^  X  -7x~*y^^='^3ox-^y*. 


■3    =-6Sa-'bH-\ 


(6)  A  polynomial  by  a  monomial : 

Multiply  each  term  of  the  multiplicand  by  the  multiplier;  add 
the  partial  products.  [th.  4 

E.g.,    {3xy^-h7y-^z*-  ^x-'z)x-ixy-^z^ 


+  |aj- 


■'y-'> 


=  _  i^x^y-^^  -  3^  xy-^z^ 

(c)  A  polynomial  by  a  polynomial : 

Multiply  each  term  of  the  multiplicand  by  each  term  of  the 

multiplier;  add  the  partial  products.  [th.  5 

E.g.,    (a^-ab-^b^)  x{a-{-b)  =  a'- a'b+  ab' -\-a'b-ab^  +  6« 

=  a^  +  b\ 


64 


PRIMARY   OPERATIONS. 


[II.  pr. 


Note  1.  Checks  :  The  work  is  tested  b}'  division,  [pr.  4 
and  sometimes  b}'  the  principles  laid  down  in  [th.  5  cr.  2-7] . 

Note  2.  Arrangement  :  The  work  is  shortened  In-  arrang- 
ing the  terms  of  both  factors,  and  of  the  product,  according  to 
the  powers  of  someone  letter  (called  the  letter  of  arrangement)  ^ 
and  b}"  grouping  together  like  partial  products. 

E.g.,  {a^  +  ^a^h-Jt^ah'-\-h^)x{a''+2ab  +  h'')  [a, let. of ar. 
is  written  a»  -f  3  a^  5  +  3  aft-  +  ft^ 

a^-{-2ab  +h- 


a«  +  3 

rt^6  +  3 

a«62_|_i 

a'W 

4-2 

+  G 

+  6 

+  2 

+  1 

+  3 

+  3 

a6* 


+  6' 


=  a^-\-ba^h   -\-\0  a^Jy" +  \Oa-V'+b  ah^  +h\ 
Note  3.    Cross-multiplication  :    The  work  is  shortened  by 
grouping  and  adding  mentally  like  partial  products,  and  writing 
their  sum  only.     E.g.,  in  the  example  of  Note  2, 
the  computer  says  :  and  writes  : 

a^    X  a^  is  a*  a* 

3a^&  X  a^  is  3a*6,  a^  x  2 ah  is  2 a* 6,  whose  sum 

is  5a^6,  6a^h 

Sab-  X  a2  is  Sa^h\  Sa^b  x  2ab  is  Ga^b\  a«  X  6' 

is  a^b-,  whose  sum  is  10  a^b',  10  a^b^ 

b^  X  a-  is  a-b\  Bab-  x  2ab  is  6a^b^,  Ba^b  X  b^ 

is  Sa^b^,  whose  sum  is  lOa^t^  10 a^b^ 

b^  X  2  ab  is  2  ab*,  3  a6^  x  6^  is  3  ab*,  whose  sum 


is  5  a6*, 
6^  X  5'  is  6^ 
and  the  whole  product,  as  above,  is 

a'  +  oa^6  +  lOa^b'  +  lOa^ft^  +  5a6*  +  6«. 
So,         to  multiply  384  by  287,     product  110208, 
the  computer  saj's : 

4x7  =  28; 
2  ;  8  X  7  =  56,  58  ;  4  x  8  =  32,  90  ; 
9;  3x7  =  21,  30;  8x8  =  64,  94;  4x2  =  8,102; 
10;  3x8  =  24,34;  8x2  =  16,50; 
5;  3x2  =  6,  11; 


5a6* 
6« 


and  writes : 
8 
0 
2 
0 
11 


3.  §11.]  PBOCESS   OF  JSnjLTIPLICATION.  55 

Note  4.  Detached  Coefficients  :  When  both  multiplicand 
and  multiplier  are  arranged  by  some  one  letter,  i.e.,  are  such 
that,  after  their  coefficients  are  detached,  the  remaining  factors 
of  successive  terms  will  stand  in  one  constant  ratio,  the  work  is 
shortened  by  the  use  of  these  detached  coefficients,  thus  : 

Take  the  terms  of  both  multiplicand  and  multiplier  in  such  order 
that,  when  the  coefficients  are  detached,  the  remaining  factors  (let- 
ters of  arrangement)  of  successive  terms  shall  have  a  constant  ratio. 

Write  the  coefficients,  suppressing  the  letters  of  arrangement, 
with  0  for  the  coefficient  of  any  term  wanting  in  either  series. 

Multiply  the  coefficients,  and  add  those  partial  products  that 
pertain  to  like  terms  of  the  final  product. 

In  the  final  product  restore  the  suppressed  factors:  in  the  first 
term  by  actual  multiplication,  and  in  the  other  terms  by  means 
of  the  constant  ratio. 

E.g.,    (a^4-3a26-f  3a62-(-6^)x(a2  4-2a6  +  Z>'), 
wherein  the  constant  ratio  of  the  literal  parts  is  6  :  a  in  both  fac- 
tors, gives  1  +  3  +  3  +  1 
1+2+  1 
1+3+  3+  1 
+  2+  6+  6+2 
+  1+  3  +  3  +  1 
1  +  5+10  +  10+5  +  1; 
and  the  product,  when  the  letters  of  arrangement  are  restored, 
is              a*  +  5a*6  +  lOd'b''  +  \Oa-b^  +  bab^  +  b'. 

Check:  1+3+3  +  1  =  8,     1+2  +  1  =  4, 

8x4  =  32      and      1+5  +  10  +  10  +  5  +  1  =  32. 

So,       16(a:«  +  2a^  +  4)  X  (a;  -  1)  +  4(a^  -  2a;  +  3) X  (x"-  3) 

gives        12      0     4  1-2      3 

1-1  10-3 


1 

2 

0 

4 

-1 

-2 

0 

-4 

1 

1 

-2 

4 

-4 

16 

16 

16 

-32 

64 

-64 

4 

-8 

0 

24 

-36 

1 

-2 

3 

-3 

6 

-9 

1 

-2 

0 

6 

-9 

4 

4-8      0    24  -36 
20     8   -32    88-100,  =  20aj*+8a^-32a;2-j-88a;-100. 


56  PRIMARY  OPERATIONS.  [II.  pr. 

So,      (ax-  Sa^x'  +aPar)  X  (b  -{-Sa^bx^-\-a*bx),  [ratio,a2a;^ 
gives        1    "3      1 

13      1  Checks:  Let  ci^a;-  =  ±l;  then, 

1    -3      i  l:f3-f  l.l±3+l=-5 

3-9      3  =1-7  +  1. 
1    -3      1 

10-701,  =abx-7a^bx^-^a^ba^.    q.e.d. 
This  method  is  a  familiar  one  in  Ai'ithmetic. 
E.g.,        1089x237  =  258,093,       or       lth  +  0h  +  8t  +  9u 
237  2h  +  3t  +  7u 

7623  7th  +  Gh  +  2t  +  3u 

3267  3tth  +  2th  +  Gh-h7t 

2178  2hth  +  ltth  +  7th  +  8h 

258093  2hth  +  5  tth  4-  8th  +  Oh  +  9 1  +  3  u 

The  first  form  is  simply  a  case  of  detached  coefficients,  wherein 

the  denominations  and  the  relations  of  the  several  numerals  are 

shown  In'  their  positions  with  reference  to  each  other  ;  as,  in  the 

last  form,  the}'  are  shown  by  words  and  signs. 

Note  5.     Type-forms  :  The  work  is  often  shortened  by  the 
use  of  certain  simple  t3'pe-forms,  which  the  reader  ma}"  prove  b}' 
actual  multiplication  and  then  memorize.      He  may  translate 
them  into  words  and  read  them  as  theorems.     They  are  : 
1]  {x-\-a)-{x-{-b)  =  x^-\-  {a  +  b)x  +  ab, 

2]  (a -\-  b) '  (a  -b)  =  d'-b^, 

3]  {a-hby  =  a^-h2ab-h^, 

4]  {a-by  =  a^-2ab  +  b^ 

5]  (a-f  6 +  C  +  . ..)'  =  «'  + 6' +  c2  +  .. . 

+  2(o6  +  acH h2>c +  ...), 

i.e.,        (2a)2    =2a-  +  S2a6, 

wherein   2a       =  the  sum  of  all  the  terms  of  the  base, 
2a-     =  the  sum  of  all  the  possible  squares, 
and  '^2ab  =  the  sum  of  all  the  possible  double  products  ; 

6]  (a  -  5)  .  (a"-i+  a^-'^b  -\-  a^-^b^-\ f-  a5"-2+  b^-'^) 

=  a**  —  6'*,     when  n  is  any  integer, 

7]  (a  +  b)  .  (a'^-i-  a^-^  +  a^-^b^ +  ab^-^-  b^-') 

=  a"  —  6",     when  n  is  any  even  integer, 

8]  (a  +  6)  .  (a"-i-  a^-^ft  +  a^'-^b^ aZ>"-2+  6''-i) 

=  a**  +  6**,     when  n  is  any  odd  integer. 


3.  §11.]  PROCESS    OF   MULTIPLICATION.  57 

E.g. ,  [(0.-^  +  f)  -h2^aff]-l{a^-\-f)-  2^x^f] 

=  x'^  +  2  x^f  +  y^-  4:aPf  [3 

=  x^-2aPf  +  y^ 

=  {^-fY.  [4 

wherein  (a^+  2/^+  2-y'a^2/^)-(a^+  ff—  2^a^y^)  is  the  same  func- 
tion of  (aj^+  y^)  and  2->/a^2/^  as  (a  +  6)  •  (a  —  6) ,  in  the  type-form, 
is  of  a  and  b;  and  later,  {oc^  +  y^Y  and  {pi?  —  'ifY  are  the  same 
functions  of  or'  and  y^  as  (a  +  2>)^  and  (a—  6)^  are  of  a  and  6. 

The  advantage  of  working  by  type-forms  is  that  most  of  the 
details  of  multiplication  are  avoided,  and  the  result  is  reached 
directly. 

Note  6.  Substitution:  The  work  is  often  shortened  by  the 
substitution,  during  its  progress,  of  a  single  letter  for  a  less 
simple  expression. 

E.g.,    to  multiply  4.a^o?  -f  96*2/2  -f-  5  -  Q>a?h^xy  -  2a^x^6 
—  Sb'y^o    by    2a^x  +  Sb'^y  + ^5. 

Let       A  =  2a^x,   3  =  36^2/?   c  =  V^, 
then        (a^  -f-  b^  -f  c^  —  ab  —  AC  —  bc)  X  (a  4-  b  +  c) 

=  A^  -f  B^  -|-  C^  —  3  ABC 

=  8a^a^ -{- 27 b^f -^5^0- ISa^b^xy  ^6. 
Note  7.     Sysimetry  :   The  work  is  often  shortened  by  noting 
the  sj'mmetry  of  the  factors. 
E.g.,    to  develop  the  product 

(2a-f  6-f  c).(a  +  26  +  c).(a-i-6  4-2c), 
write  the  factors  in  three  lines, 
2a+    b+    c 
a-\-2b+    c 
a-\-    b-\-2c 

then  *.•   the  product  has  the  terms         2a*  a  *  a,  =  2 a^,   [th.5 
2a.  a  •  b  +2a-2b'  a  -\-b  •  a  -  a,=  la^b, 
2a-2b'2c-\-   b  -  c  •  a  -i-c  -  a  •  b 
+  2a-  c  •  b  -\-   b  •  a  .2c+c  .26- a,  =16a&c; 

and    • .  •    every  term  of  the  product,  being  entire  and  of  the  third 
degree,  is  of  like  form  to  one  of  these  as  to  a,b,c; 

and    • .  •    the  product  is  a  symmetric  function  of  a,  &,  c  ;  [th.  5  cr.2 


58  PREMARY  OPERATIONS.  [II.  pr. 

.*.    it  has  likewise  the  following  terms,  and  no  others  : 
26^  2  c^  as  well  as  2a^ 

and  Ih^c^  Ic^a,  7a6^  7b(f,  7ca^  as  well  as  7a^b; 
.-.    (2a  +  b-\-c)-{a  +  2b-\-c)'\^a-\-b  +  2c) 

=  2a«+26»  +  2c3-}-7a26H-76-c  +  7c2a 

+  7a62-f-  7bc^  +  7ca^  +  lQabc 
=  2Sa3  +  72a-6  +  16a&c; 
wherein    Sa^  =  the  sum  of  all  the  possible  cubes, 
and  2a^6=  the  sum  of  all  the  possible  products  got  bj^  tak- 

ing one  letter  twice  and  another  letter  once. 
Check:  The  sum  of  the  coefficients  in  each  factor  is  4, 
and  in  the  product  it  is  64,  =4x4x4.  [th.  5  cr.  4 

So  •.•  of  (2a  +  6  — c).(— a  +  26+c).(a-6  +  2c), 
the  terms  in  a^,  b^,  c^  have  the  same  coefficient,  —  2, 
those  in  a^b,  b'c^  (?a  have  the  same  coefficient,  5, 
those  in  a6^,  6c^,  cc^    have  the  same  coefficient,  —  1 , 

and  that  in  abc  has  the  coefficient  2 ; 

.•.-2(a»+&8+c')  +  5(a26+&'c+c2a)-(a624-5c2+ca2)-f2a6c, 
=  — 22a3  +  oSa^ft  —  %ab^  +  2a5c,  is  the  product. 

Check  :   The  sum  of  the  coefficients,  when  the  brackets  are 
removed,  is  8,  =  (2  +  1  -  1)^  [th.  5  or.  4 

So,  to  develop  the  sum 

(a  +  6  -  2 c) 2  +  (6  +  c  -  2 a)2  +  (c  +  a  -  2  &) 2, 
get  by  multiplication ,  or  from  the  type-form  for  (a + 6  H —  )  2,     [5 

(a  _|-  5  _  2c)2  =  a2  +  52  _j.  4^  _f.  2a&  -  4ca  -  4&C, 
write,  by  symmetry, 

(6  +  c-2a)2  =  62  4.c2  4.4a2+26c-4a&-4ca, 

(c4-a-26)2  =  c2+a'4-462-f-2ca-46c-4a&, 
and  add ;  the  result  is 

6(a2  +  62  +  c2_&c-ca-a&),  =^{^a^ -^ab). 
Check  :    As  Sa^,  Sa&,  each  have  three  terms,  their  coefficients 
in  the  sum  of  the  three  products  are  the  sum  of  coefficients  of 
a^,  •••,  and  of  a5,  •••,  in  the  product  first  got ;  i.e.,  6  and  —  6. 

In  such  sj'mmetric  expressions,  where  three  letters  are  in- 
volved, the}^  may  be  kept  advancing  in  the  same  order, 

abc^  bca,  cab  or  acb,  bac,  cab      ab,  be,  ca  or  ac,  ba,  c&, 


3.  §11.]  PEOCESS   OF  MULTIPLICATION.  59 

as  if  they  were  points  on  a  circle  following  one  another  round 
and  round  in  the  same  rotary  direction. 

So,       {a+b+c)'(x+y+z)  +  {a+b-c)'{x+ij-z) 

+  (a-b+c)'{x-y+z)  +  (-a+b+c)-{-x+y+z) 
= -{- ax -^  ay  -\- az -\- bx -[- by  +  bz -{- ex -{•  cy  -{- cz 
+       +       -      +      +       ---      + 
+       -+-      +       -      +      -      + 
+       ---      +      +-      +      + 
=  4aaj  +45y  +4:cz 

=  4  S  ax,  a  symmetric  expression  as  to  ax,  by,  and  cz. 

Note  8.  Contraction  :  When  only  the  first  few  terms  of  a 
product  are  wanted,  the  work  may  be  shortened  by  omitting  all 
partial  products  that  do  not  enter  into  the  required  terms. 

E.g.,    to  develop  (1— 3a;H-5a:^ )^as  far  as  the  term  in  a^: 

l_3a;-f  5ar^ or     1-3+  5 

1-30;+  5a^ i_34,  5 

l^Sx-h  5x^  1-3+  5 

-3a;+  dx^  —3+  9 

+  5a^  +  5 

l-6a;  +  19a^  1_6+19 

So,       to  find  the  product,  omitting  x*  and  higher  powers,  of 

(l^x-{-x'-^...)'{l-2x+Sx' ).(l+4a5+9.T2+--.), 

write 


1    1 

-2 

11X1 

-2  -2 

3     3 

-4 

-2     3  -4 

1  -1 
4 

2-2x1 

-4     8 

9  -9 

16 

4     9  16 

1     3     7  13, 
and  the  product,  as  far  as  wanted,  is 
l+Sx  +  7x'+lSa^. 

If  in  this  example  the  value  of  x  be  small,  say  a;-^  .01,  the 
part  product  found  above  approximates  very  closely  to  the  value 
of  the  true  product,  and  for  many  purposes  ma}^  be  used  in  place 
of  it.     The  work  is  further  shortened  by  cross-multiplication. 


60  PRESLVKY  OPERATIONS.  [II.  pr. 

This  method  of  contracted  multiplication  may  be  used,  with 
great  profit,  with  decimal  fractions. 

E.g.,,    to  find  the  product  37.8562  x  14.9716,  correct  to  two 
places,  and  .2819  x  .3781  x  .2148  to  three  places. 


37.8562    ar 

id    .2819 

.107 

14.9716 

.3782 

.2148 

378.562 

.0846 

.0214 

151.425 

197 

11 

34.070 

22 

5 

2.650 

.107 

.023 

38 

23 

566.77 
In  writing  down  the  partial  products,  cany  what  would  have 
been  carried  had  the  multiplication  been  made  in  full. 

E.g.,    the  partial  product  23  =  3  X  6  +  5,  carried  from  8x6. 

§12.     PROCESS   OF   DIVISION. 

PrOB.  4.       To  DIVIDE  ONE  NUMBER  BY  ANOTHER. 

(a)  A  monomial  by  a  monomial : 

To  the  quotient  of  the  numerical  coefficients  annex  the  several 
literal  factors,  each  talcen  as  many  times  as  its  exponent  in  the 
dividend  exceeds  its  exponent  in  the  divisor.  [th.  3  cr.  9 

Mark  the  quotient  -{  _  if  the  terms  are  of  ■{  Jf.Qy,f^ny.y  sense. 

E.g.,        6Sa-^bH'     :      7ac^d'      =      da-^b^'c-^ 
—  Sox^y~*!i*      :      oocy'^z^    =—7a^y~^z~^y 

^a-'b-'d-' :  -  -ac-^d-'  =  -  -a-^ft" V, 
25  5  5 

3       ^  2       ^  21  ^ 

(b)  A  polynomial  by  a  monomial : 

Divide  each  term  of  the  dividend  by  the  divisor;  add  the  partial 
quotients.  [th.  4 

=  -no?^-'^y-'z  +  llx-'y-'z-\ 


4.  §12.] 


PROCESS  OP  DIVISIOIT. 


61 


(c)  Any  number  by  a  polynomial : 

Arrange  the  terms  of  both  dividend  and  divisor  according  to 
the  poivers  of  some  one  letter,  (preferably  the  letter  whose  powers 
are  most  numerous,  aucl  the  highest  powers  first). 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  called  the  trial  divisor. 

Midtiply  the  ivhole  divisor  by  this  partial  quotient,  and  subtract 
the  product  frmii  the  dividend. 

Repeat  the  work  upon  the  remainder  as  a  new  dividend,  and 
so  on,  till  the  whole  dividend  is  exhausted,  or  till  the  requirements 
of  the  icork  are  satisfied. 

Add  the  partial  quotients :  their  sum  is  the  quotient,  the  part 
of  the  dividend  left  undivided  is  the  remainder,  and  the  sum  of 

the  quotient  and  the  fraction  — :^^—^ is  the  complete  quotient. 


divisor 
E.g. ,    to  divide  a^  +  6^  by  a  +  6. 


a^+b^ 

a  -i-b 

a^-\-a^b 

a'-ab-{-b^ 

-d'b  +  b^ 
-a'b-ab^ 

ab^-{-b^ 
ab'-^  W 

or 


a2_a6  +  &2 
a  -\-b 

ab'' 


quotient^ 

divisor, 

dividend. 


The  first  is  the  ordinar}^  form  for  division  ;  and  the  other  is  an 
abbreviated  form,  in  which  the  partial  products  are  not  written, 
but  the  partial  remainders  only.  The  relative  position  of  divi- 
dend, divisor,  and  quotient,  is  unimportant  except  as  a  matter 
of  convenience. 

So      (ar''—a)  :  (r  —  1)  =  ar"^^+  ar^'-^^  ar'^-^-\ \-ar'{-a, 

and         (a  —  ar"")  :  (l  —  r)  =  a  +  ar  +  ar^-\ f-  ar"-2+  ar"~^ 


So 


a^-h2aa^-\-a^ 


dividend 


+  a& 


x-{-2a^b 


X  -\-a 


divisor, 


-\-b 


x-{-ab 
+  2b^ 


quotient, 


a 

+  b 


7?       ab 
,4-26^ 


a^b 
-ab^ 
+  2b^ 


remainder. 


This  example  is  solved  by  the  second  method,  and  the  several 


62  PKIMAEY  OPERATIONS.  [II.  pr. 

partial  remainders  are  written  on  one  horizontal  line  simply  as  a 
matter  of  convenience.     The  complete  quotient  is 


QC--\-a 


x  +  ab  I  ,  a^b-ab^-\-2i 
+  2b'r      x-\-a-b 


Note  1 .  Checks  :  The  work  is  tested  by  multiplying  together 
the  divisor  and  quotient,  and  adding  in  the  remainder.  In  this 
process  the  order  of  multiplication,  quotient  X  divisor,  is  prefer- 
able, as  least  like  the  order  of  the  work  of  division. 

K  the  division  be  exact,  principles  laid  down  in  [th.  5  crs.  2-7] 
are  also  useful. 

Another  test  is  to  reproduce  the  divisor  b}^  subtracting  the 
remainder  from  the  dividend  and  dividing  by  the  quotient. 

Note  2.  Different  forms  of  Quotient.  Unless  the  division 
is  excwt,  i.e.,  leaves  no  remainder,  the  quotient  and  remainder 
are  commonly  different  for  every  different  choice  of  trial  divisor 
that  may  be  made. 

E.g..,  (a^+  1)  :  (»  +  1)  gives  for  quotient  and  remainder : 
a;  —  1,  2       when  x  is  trial  divisor, 
1  —  a;,  2a^  when  1  is  trial  divisor. 

So      (ic^+ 2/^4-2^)  :  (ic+y+z)  gives  for  quotient  and  remainder: 
x  —y  —  Zy   2(y--^yz  -{-  z^)  when  x  is  trial  divisor, 
y  —z  —x^  2(z^-^zx  -^x^)  when  y  is  trial  divisor, 
z  —  x  —  y^  2{x^-\-xy  -{-y^)  when  z  is  trial  divisor. 

The  quotient  and  remainder  depend  also  upon  the  extent  to 
which  the  division  is  carried. 

E.g.,  (a^+  1)  :  (x  +  l)  gives  for  quotient  and  remainder : 


X,        -x  +  1 

or 

x-1,                  2 

or 

x  —  l-\-2x-^,  —2x-^ 

or 



a^+1 

—  x-\-\ 

2 


x-\-\ 

x  —  \-\-2x-^—2x-^ 


But  the  complete  quotient  is  the  same  in  value,  whatever  the 
trial  divisor  and  the  extent  of  the  division. 

E.g.,    X-] J^  =  x-l-\ —  =  x-l+2x-^-] rT-='" 

l-\-  X  2aP 

=  1  -} ^^^—  =  1  —X -\ =  ....  when  1  -}- a;  is  divisor. 

1+a;  1-faj 


4.  §  12.] 


PEOCESS   OF  DIVISION. 


63 


Note  3.  Detached  Coefficients  :  When  both  dividend  and 
divisor  are  arranged  by  some  one  letter^  i.e.,  are  such  that,  after 
the  coefBcients  are  detached,  the  remaining  factors  of  snecessive 
terms  will  stand  in  one  constant  ratio,  the  work  is  shortened  by 
the  use  of  these  detached  coefficients,  thus  : 

Take  the  terms  of  both  dividend  and  divisor  in  such  order  that, 
when  the  coefficients  are  detached,  the  remaining  factors  of  suc- 
cessive terms  shall  have  a  constant  ratio. 

Write  the  coefficients,  sujipressing  the  letters  of  arrangement, 
with  0  for  the  coefficient  of  any  term  wanting  in  either  series. 

Divide,  treating  the  set  of  coefficients  in  the  dividend,  and  that 
in  the  divisor,  as  polynomials. 

In  the  quotient  restore  the  suppressed  factors:  in  the  first  term, 
by  actual  division,  and  in  the  successive  terms,  by  means  of  the 
constant  ratio. 

E.g. ,  (a^  -h  W)  :  (a  -h  6)  and     (a^  -  b^)  :  (a  +  &) 


give 


0    1 


1    1  and       10    0-1 

1  -1    1  -1,  1,-2 


1    1 


1  -1    1 


-1 
-1 


1    1 
and  the  quotients  and  remainder  are 

a^  —  ab-{-  W   and   ci?  —  ab  4-  6^, 


2hK 


So 
gives 


1-2    0    6-9 
-2    3 


1    0-3 


1-2    3 


3    0 


and  the  exact  quotient  is 
a^  — 2a;  +  3. 

Note  4.  Synthetic  Division  :  When  the  first  coefficient  of 
the  divisor  is  1,  the  work  by  detached  coefficients  is  further 
shortened,  thus : 

Suppress  the  first  coefficient  of  the  divisor,  and  replace  the  other 
coefficients  by  their  opposites,  so  that  the  partial  products  may  be 
added;  write  the  skeleton  divisor  thus  changed  preferably  at  the 
left  in  a  vertical  column  running  dowii,  the  partial  products  under 


64  PKIMAllY   OPEEATIONS.  [II.  pr. 

the  dividend^  and  the  quotient  under  these  partial  products;  add 
the  partial  products  as  needed. 

E.g.,    (a;^  +  3ar»4-3ic2+2):  (ar'-2a;-f-3) 
gives 


1 

3 

3 

0 

2 

2 

2 

10 

20 

-3 

-3 

-15 

-30 

1 

5 

10, 

5 

-28 

and  the  quotient  and  remainder  are 

ic^  +  Sx  +  lO,   and   5a;— 28. 
Note  5.     Type-forms:   If  the  dividend  and  divisor  can  be 
reduced  to  known  type-forms,  e.g.  [1-8] ,  then  the  quotient  may 
be  written  directly. 

E.g.,{o?-\-lx  +  U):{x-\-^)^x  +  ^,  [1 

{x^-f)  :  {x-\-y)=x-y,  [2 

{:>^-f)  :  {x^-f)  =  aP  +  x*f  +  a^7/+f,  [6 

wherein   a^  =  a,  y^  =  b,  4  =  n,    of  the  type-form. 

Note  6.  Substitution  :  The  work  is  often  shortened  by  the 
substitution,  during  its  progress,  of  a  single  letter  for  a  less 
simple  expression. 

E.g.,  (8aV  +  276V  +  5V5-lS«^&'a;2/V5) 
:{2a'x  +  3b'y  +  ^5) 
when        A  =  2a^x,    B  =  Sb^y,   c=V^, 
gives      (A^-f  B^  +  c^  — 3abc)  :  (a  +  b  +  o) 

=  A^  4-  B^  +  C^  —  AB  —  AC  —  BC 

==Aa^x--\-db'y^-\-5-6a^b-xy-2a^x^5-Sb^y^5. 

Note  7.  Symmetry  :  If  the  dividend  and  divisor  be  both  sym- 
metric as  to  two  or  more  letters,  and  if  there  be  no  remainder, 
then  the  quotient  will  be  a  symmetric  function  of  the  same  let- 
ters. It  is  then  often  sufficient  to  get  a  few  characteristic  terms, 
and  to  write  the  rest  therefrom  by  sj^mmetry. 

E.g.,  to  divide  (a^ +b- -\- (^ -}- 2 ab  +  2 be  +  2 ca)  by  (a+b+c), 
wherein  both  elements  are  S3'mmetric  functions  of  a,  b,  c. 

Manifestly  a  is  the  first  term  of  the  quotient, 

.  • .  b  and  c  are  also  probable  terms  of  the  quotient ; 
and     •.•  the  product  (a4-6H-c)-(a-}-6+c)  is  the  given  dividend, 
.-.  the  division  is  complete,  and  a  +  6  +  c  is  the  quotient. 


5.  §  13.]  OPERATIONS   ON  FRACTIONS.  6o 

So      (a^y  +  xy--\-2fz-\-  yz^+  z^x  +  zx^+^  xyz)  :  (x-\-y  +  z), 
a  quotient  of  symmetric  fmictions  of  ic,  y^  and  2;, 
gives       xy  for  one  term  of  the  quotient ; 

.'.  yz  and  zx  are  also  probable  terms  of  the  quotient ; 
and    • .  •  the  product  {x-\-y+z)  {xy-{-yz-\-zx)  is  the  given  dividend, 
.*.  the  division  is  complete,  and  xy-\-yz-\-zx  is  the  quo- 
tient sought. 
Had  the  last  term  of  the  dividend  been  4:Xyz^  or  any  other 
number  except  3  xyz^  there  would  have  been  a  remainder.     The 
reader  must  therefore  use  great  caution  if  he  employs  "  sym- 
metry'" in  division.     He  may  safely  use  it  as  suggestive  of  the 
true  answer,  but  hardly  ever  as  conclusive. 

Note  8.  Contraction  :  When  only  the  first  few  terms  of  a 
quotient  are  wanted,  the  work  is  shortened  by  omitting  all  par- 
tial products  that  do  not  affect  the  required  terms. 

E.g.,    (l+aj  +  a^  +  a.-^-f-...)  :  (i  _  2a;  +  3a^-4a^-f-...) 
to  four  terms 
gives 


1 

1 

1 

1 

2 

2 

6 

8 

3 
4 

-3 

-9 
4 

1 

3 

4 

4 

[nt.  4 

and  the  quotient,  as  far  as  wanted,  is 
l+3a;  +  4a;2  +  4a;3. 

§  13.      OPERATIONS  ON  FRACTIONS. 
PrOB.  5.       To  REDUCE  A  FRACTION  TO  LOWER  TERMS. 

Divide  both  terms  by  any  same  number  that  divides  them  without 
a  remainder;  the  quotients  are  the  terms  of  the  reduced  fraction. 
^         36a*5V     ZM  [th.  3  cr.  7,  cr.  5  tit. 

^        24a^to       2ax 
wherein  the  divisor  is  12a^6  ;  and  this  common  operator  may  be 
written  under  the  sign  = ,  so  that  the  whole  stands  in  the  form 
36a^6^c^   =    5^. 
24  a*  6a;  {naU)  2  ax 

Note.     For  reduction  to  lowest  terms,  see 


66  PEEMARY  OPERATIONS.  [II.  pr. 

PrOB.  6.  To  REDUCE  A  FRACTION  TO  A  GIVEN  NEW  DENOMINA- 
TOR OR  NUMERATOR. 

Divide  the  new  denominator  or  numerator  by  the  old,  and 
multiply  both  terms  of  the  fraction  by  the  quotient. 

[th.  3  cr.  7,  cr.  5  nt. 

E.g.,    to  reduce  — ^  to  an  equivalent  fraction  whose  denom- 
inator is  a^bc: 
•r   a^bc  :  2a^b  =  iac, 
Ss^y  _  ^ac3?y 
2a^b  (^)    a^bc 

2a^2 
So        to  reduce  — —  to  an  equivalent  fraction  whose  numer- 
Sa^c 

ator  is  Ga^yz: 
-,'    Gs?yz  :  2a^z  =  Sy, 
2^z  _  G^yz 
3a-c  (3y)  ^d^cy 

Note.  B}^  this  rule  any  entire  or  mixed  number  is  reduced  to 
a  simple  fraction. 

fP  n      r.  I  ^„_x-^2a_dx-\-2ad 
1  a 

So        x-\-2a-\ —  =  — ■ ■ 

X  X 

PrOB.  7.  To  REDUCE  TWO  OR  MORE  FRACTIONS  TO  A  COMMON 
DENOMINATOR. 

Over  the  continued  prodicct  of  the  denominators,  write  the  prod- 
uct of  each  numerator  into  all  the  denominators  except  its  own. 

[th.  3  cr.  7,  cr.  5  nt. 
■pj         5xy    Sbc    3(a— 6)  _  SoaFy    4:2abc     6 ax  (a  —  b) 
2a'     a;  '  7  14aa;'    14aa;'         14 aa; 

Note.  The  fractions  may  be  reduced  by  Pr.  6  to  any  common 
denominator  whatever ;  but  this  usually  leads  to  complex  frac- 
tions, which  the  rule  of  Pr.  7  avoids  when  the  given  fractions 
are  simple. 

For  reduction  to  lowest  common  denominator,  see  III.  §  6. 


G-9.  §  U.]  EXAI^IPLES.  67 

PrOB.  8.       To  ADD  FRACTIONS. 

Reduce  the  several  fractions  to  a  common  denominator^  and 
write  the  sum  of  the  new  numerators  over  the  common  deihomi- 
nator. 

^         Sbc^     3(a-6)^216c^  +  6aa;(a-6) 
2  ax  i  \4:ax 

Note.    Subtraction  is  but  a  case  of  addition  ;  add  the  opposite         ^ 
of  the  subtrahend. 

^         3bc^     B(a^b)  ^21b(^-6ax(a-b) 
'^''    2  ax  7  14  aa; 

Prob.  9.     To  multiply  fractions. 

Write  the  product  of  the  numerators  over  the  product  of  the 
denominators.  [th.  3  cr.  1 , 3 

'^*'    2aa;  7  Uax 

Note.  Division  is  but  a  case  of  multiplication ;  multiply  by 
the  reciprocal  of  the  divisor. 

'^''    2ax  '         7  2ax      3{a-b)      2ax{a-b) 

§  14.     EXAMPLES. 
§§  9,  10.     PROBS.  1,  2. 

1-8.   Free  from  brackets  and  reduce  to  simplest  form  : 
(a)  removing  first  the  inner  brackets,  and  proceeding  outwards  ; 
{b)  removing  first  the  outer  brackets,  and  proceeding  inwards  ; 
(c)  freeing  together  all  terms  of  a  kind,  from  all  the  brackets. 

1.  a-[6-(c-d)]. 

2.  a-\a  +  b-[a-\-b-c-(a-b-\-c)~\\, 

3.  _^(l  +  2a;  +  9a^)+[(3  +  2a;-.'c2) 

-  (2  +  5a;  +  7a.-2)  +  (- 3  +  3a;  -  2a^)] ^. 

4.  |[|(a;-a)  +  (2/-6)]  +  |[2(x-a)4-i(a-a^)  +  2(2/-&)]. 

5.  -iJ[(5a-464-3c)-(-3a  +  46-c)] 

-[(6a-8c)-(a-6  +  9c)]5. 

6 .  ^{ax'-\-bx  +  c)-\l {ax^ -bx-{-c)  +  ^{^ao?  +  bx-\c) 

+  i(-aa;2  +  2Z>a;  +  c)]. 


68  PKEMARY  OPERATIONS.  [II. 

7.  mx^-f)-{7?+2xy+f)-]-\_{2xy-;^-f)-\{a?+f)-]. 

8.  1. 25  [1.12a;-. 24  (ic-. 5)]  

-^[.21(a;  +  l)-.lo(l-.lGa;)-.12(a;-l-.25a;)]. 

9.  Add  (a-2i))a;3  +  (g-6)iB2-f  (3c-2r)a;  +  (3i)-a)aj» 

-{2Q?-x)-{c-l)x-{h  +  q)a?-{p-a)x? 
-iB»  +  3  6ar^-(c-2r)a;, 
and  arrange  the  sum  to  ascending  powers  of  x. 

10.  Arrange  a^^  6^4-  0^+  3 0^6  +  3 6-c  +  3 c^^  _^  3 ^2)-  +  3 60^ 

+  3ca2+6a6c 
(a)  to  ascending  powers  of  a,  using  vertical  bars, 
(6)  to  ascending  powers  of  6,  using  horizontal  bars, 
(c)   to  ascending  powers  of  c,  using  brackets. 

11.  XMx'^-\-^xf-x:^^7?y  +  Q^z  +  Za?y'^-it^x'z--\-Zxy^z 

—  3  xyz^—  Gxryz  —  if^2/  +  2/*  ~~  2/^—  3  xry-—  3  a;?/^2J 

—  3  a/- 3 02/22- 3 2/3^+ 3 2/2;22_  g 3.^2^  _  ^2  ^  3 2^ 2; 

+  ;2*  +  3  iB22,2_3  a:2^2^3  a^^  _|.  3.23_  3  2,2^2  _^  2/;z3  ^  3  a^^2^ 

and  arrange  the  sum  to  descending  powers  of  jc,  and  the 
coefficients  to  descending  powers  of  y. 

12.  From  a'^— 4  a^b^— S  a^ir^— 17  ab*— 12  b'^,  subtract  successively 

a»_2a*6-3a352,   2a*b-4:a^b''-6a^b\ 
Sa^b^-Qa^b^-dab*,   and  4.a'b^-8ab*- 12b''. 

13.  Jf     s  =  {a-{-b+c)x-{-{a+b-j-c)y,  v  ={b-\-c)x-{-(2b-c)y, 

v=(c  +  a)a;  +  (2c  — a)2/,  and  ■w={a-\-b)x-\-(2a—b)y; 
find  the  values  of  (s  —  u)  +  (v  —  w) ,  (s— v)  +  (w— u) ,  and 
(s  — w)  +  (u— v) ,  and  the  sum  of  these  three  sums. 

14.  Express  b}^  brackets,  each  preceded  b}'  +  ;    each,  by  — "; 

each  beginning  with  a  -f  term  ;  taking  the  terms 
(a)  two  together,  in  their  order, 

(6)  three  together,  with  an  inner  bracket  embracing  the  last 
two  of  each  triplet : 

—  3c  +  4d  — 2e+3/  +  2a  — 5&; 

—  2e4-3/  +  2a-56  — 3c-4d; 
2a-56-3c-4d!-2e+3/; 

a  +  b-{-c  —  a  —  b-\-c  +  a  —  b  —  c^a-{-b  —  c; 

abc  —  abd  +  abe  —  acd  +  ace  —  bed  +  bee  —  bde  4-  ade. 


§14.] 


EXAMPLES. 


IS- 


IS. 
16. 
17. 
18. 


19. 

20. 

21. 
22. 

23. 
24. 
25. 

26. 
27. 
28. 

29. 
30. 
31. 
32. 

33. 
34. 
35. 


79.   Multiply  and  divide  as  shown  by  the  signs;   use  the 
methods  given  in  the  problems  and  notes  specified : 

§  11.     PROB.  3,  and  ?jote  3. 
(a;3_  2a^_  3a;  +  1) .  (2a^- 3ic  +  4) . 

(a.-s-f  ax  -  b^) .  {x^-\-  bx  -  a")  •  {x  -  IT^b) . 
[(a-l)a^+(a-l)V+(a-iyaj].[(a+l)a;  +  (a+l)2 

^{a-i-iyx-'^. 
Use  vertical  bars  to  join  coefficients  of  like  powers  of  x  in 

the  product. 
(a'«4-36~-2c^).(a-"'— 3&-"+2c--p). 


x^-{-a 


x  +  ab  '  aj^+c 


x-\-cd 


-b 


x-\-ab  •  a^+c 


x-\-cd. 


ar+a 
+  b 


x-^ab  -  ocF—a 
-b 


x-\-ab',    x'^—a\x-\-ab  '  ay^—c 
-b\  -d 


x-\-cd. 


(x -\- a) '  (x -{- b) '  (x -\- c)  •  {x -\- d) ,  at  one  operation. 

§  11.      PROB.  3,  NOTE  4. 

(3^  -  Sa^y^  +  3xy*  - y^)'{x^ -  Ax^y^  -{-  Qx^y^ -  4:xy^  +  f) . 

(^a^-2x^-\-l)-{2x^-Sx  +  4:)'{x  +  l). 

(a^  —  mx  4-  w^)  •  {a^  +  mx  -\-  m?)  •  (a;*  +  m^x^  -{-  m*) . 

Show  that  a; .  (a;  + 1)  •  (a;  +  2)  •  (a;  +  3)  +  1  =  (a;^  +  3 a;  +  1)2. 

Showthat(2/-l).2/.(2/  +  l).(2/  +  2)  +  l  =  (/  +  2/-l)'. 
What  function  must  a;  be  of  2/  so  that  Exs.  26  and  27  shall 

be  precisely  the  same  equation  ? 
(a/  4-  &?/V  —  cy'^z^)  •  {ay^z^  —  bif^  +  cy^) . 
(2a;  +  3).(3aj-4);   (3^/- 5).(22/+ 7). 
(a^  +  3a;  +  2).(a;2_3a;  +  2);   (2 -4/).(H-22/2). 

(a^  +  3a;2  2/ +  3a;/ +  2/3) .(a;2  +  2a;2/ +  2/")  ; 
(2a;8  -  3aj2  2/ +  22/3) . (2a;3  _^  3a.2^2  ^  22^). 

(2a; -5)2;   (7/ -1-22/2  + 32/3)2;   (2  -  3;<;- 3^2  + 2;s3)2. 

13  X  15  ;  35  X  79  ;  234  X  432  ;  135.7  X  12.34. 

182;  372;  ^gg^ ;  1632;  7252;  18812;   70.232. 


70  PKIIVIARY  OPERATIONS.  [II. 


36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 
44. 
45. 
46. 
47. 
48. 
49. 


51. 
52. 
53. 
54. 


55. 
56. 
57. 

58. 
59. 


§  11.      PROB.  3,  NOTE  5. 

x-\-2y{x+3);  (x+2y{x-Sy,  (x-2y{x+3)',  (a;-2).(a;-3). 

y+a)'{y+b);  {y^ay{y-h);  (y-.a).{y+by,(y-{-a)iy-b). 

x-\-a  +  b)'(x-{-c-^d);   {x  —  a+l)) -{x  —  c-^d). 

3a2iB3  +  56V).(3aV  +  5  6V). 

ax'  +  f)'(ax'-f);   {2a^  +  3f'z)'{2a^ -3y^z). 

m'zi  -  n'yi)  •  {mH^  +  n'y^)  ;   (2  +  V^)  •  (2  -  V^)  • 

x-'a)'{x  +  a)',  (x^-a^).(oi^+a^);  ... ;  (a;"-a'*).(a;'»+a"). 

I_a;).(l+ic.).(l+a^).(l+a;^).(l4-fl^)...(l4-a;2«). 

x-\-3yy;  (x-Si/Y;   (^^  ±  3)^   (2a^±3/)2. 

a  +  6ir3)2;   (a-F+c)2;   (F+6±c"=^)^ 

x-\-y-hzy;   {2x-h3y^Azy;  (xy +  yz-{-zxy. 

a  +  6  +  c)-(  — a  +  &  +  c)  •(«-&  + c).  (a  4-&  —  C). 

a4-26  — 3c  — d).(a-26  +  3c-d). 

a-{-mx  —  ns?)  '{a  —  mx-\-  ns?) . 


50.  fx+-^_^vri--+— Y 

V        2a      4aV    V       2a      4aV 


a?  -\- ax -^  a?) '  {a  —  x)  X  (a^  —  ax  +  x^)  -{a-^-x). 

af~^  —  x''-^y-\-x'*-^y^ ±  ic^''""  T  2/""^)  •  (^  +  2/)  • 

p -\- pr  +  p-r^  -\- pr^ -^ \- pf^)  -  {r  —  1)  . 

§  11.      PROB.  3,  NOTE  6. 


u  +  v  +  x  -Fir)-  +  (?^  -h  V  -  «  +  2/)^- 
w  —  v  +  a;  —  yY  —  {u  —  v  —  x  —  yy. 


a  +  b  ■\-3a-{-b'm +  71 -\-m  +n)  '  {a -{-b  —3a -\-b'm-\-n 

+  W14-71  ) 

a^r6  +  3ar+^.m  +  n^  +  m  +  7i).(a+3-3a+6^.m^^ 
4-  m  -h  w) . 

V2  +  V3  +  V5)-(- V2  +  V3  +V5)-(V2  -  V3  +V5) 
.(V2+V3-V5). 


§  11.]  EXAMPLES.  71 

§  11.      PROB.  3,  NOTE  7. 

60.  (ah  +  cdy  +  (be  +  ady+  {ca  +  bdy=:  ta^b^-}-  6 abed, 

61.  {ax  —  byy-i-(bx  —  cyy+{cx  —  ayy 

=  {ay  —  bxy-\-  {by  —  cxf  +  {cy  —  axy. 

62.  {-x  +  y  +  z){x-y-{-z){x  +  y-z)  =  :^a^y-^a^, 

63.  {a  +  b)-{c  +  d)  +  {a-{-b)'{c-d)-{-{a-b)-{c  +  d) 

+  (a-6).(c-d). 

64.  (ax  +  by)  •  (5a;  +  ay)  -\-  {ax  —  by)  •  {bx  —  o?/) . 

65.  (a;  +  2/'f^)'-(aJ^  +  /  +  2')  =  3(a;  +  2/)-(2/  +  ^)-(2!  +  a;). 

66.  {a-\-b  +  cy  =  ^a^  +  3%an-{'6abc. 

67.  If  2a  =  a  +  6  4-c  +  ---,  '      ' 
then  (2a)3    =  Sa^  +  2a'b-{-6 %abc, 

2a.2a&=2a26  +  32a&c. 

68.  {-a  +  b->rC-\-d)'{a  —  b  +  c-\-d)'{a-\-b  —  c-\-d)'{a-{-b 

+  c  -  d)  =  — 2a^  +  2  2a2  62  +  8  a6cc?. 

69.  (aa;  +  by  -\-  cz)  •  (6a;  +  c?/  +  az)  •  (ca;  +  a?/  +  bz) 

—  abc  {x^  +  y^  ■\-z^)  +  {a^  +  b^  +  c^)  a;?/2;  +  3  a&c  •  xyz 
+  (a62  _|.  5c2  +  ca^)  (a;?/^  +  yz"  +  i^a;^) 
+  (a26  +  6-c  +  c2ci)  (ai2?/4-/;2  +  ;s2  3,>)^ 
Test  the  result  severally  by  [th.  5  crs.  2-7]. 

§  11.      PROB.  3,  NOTE  8. 

70.  (l-|a;4-iaJ^-fa;2  4-...)2  to  four  terms. 

71.  (1 +  . 167a; +. 014 a;2_|_  001  a;3)2 

X  (1  -  .333a;  +  .056a;2  _  .0063;^+  •.•)• 
Carry  nothing  beyond  the  third  decimal  place,  and  retain 
nothing  beyond  the  term  in  ^. 

§  12.      PROB.  4. 

72.  3a26:a6;    -3aa;:-a;2;  ^^-^r-m^n;   -T^st-'^ilr-^^f^ 

73.  (ar2  +  2aa;  +  6):a;;    (Ja;'-|a;2/-'  +  f  r')  ^  -  3a^r'- 

74.  (2/2 +  52/+ 6):  (2/ +  2);    (15a;«+a;2^-i_j_4y-3)  .  (3^_j  22/-i> 
76.  (a"*+ "  —  a"^ 6"  +  a" 5"*  —  6"'+ ") :  (a"  —  6**) . 


72 


PRIMARY   OPERATIONS. 


[II. 


76.   x*  +  a 

x"  +  ah 

3?  +  ahc 

+  b 

-f-ac 

+  ahd 

+  c 

-\-ad 

-\-acd 

+  d 

-\-hc 

+  hcd 

+  bd 

+  ccZ 

X  +  ahcd  -^  mr  -\-a 


x  +  ah. 


§  12.      PROB.  4,  NOTE  2. 

n.  x:{x-\-a),  {a-\-x):{h  +  x),  a^ -,  {a -\' xY ,  a:(l+ic), 
(1 -f  2a;):(l  —  3a;),  1  :  (1  —  2a;  +  or^),  each  to  four 
places  of  the  quotient,  then  write  the  complete  quotients 
b}'  annexing  the  remainders  written  over  the  divisors. 

78.  (1  +  a;^— 82/^+6a;?/)  :  (1  +  a;— 2?/),  first  to  ascending  powers 

of  a;,  second  to  descending  powers  of  y. 
Test  b}'  multiplication. 

79.  {lSxyZ'\-'21^-a?-[-Sf):(x-^z-2y). 

Test  severally  by  all  the  principles  in  [th.  5.  crs.  2-7]. 

80.  Show  that  if  x^  +px  -\-qhQ  divided  by  x  —  a^  the  remainder 

is  the  value  the  dividend  has  when  a  is  substituted  for  x. 

81.  So,  if  x^  -hpxr  -f  ga;  +  r  be  divided  by  x  —  a. 

82.  So,  if  X*  -{-pa^  -f-  qa:r-{-  ra;  4-  s  be  divided  by  a;  —  a. 

§  12.      PROB.  4,  NOTES  3,  4. 

83-86.  Divide  by  detached  coefficients,  and  b}-  synthetic  division  : 


83.  ar^-9a;2_j_26a;-24  :  a;-4;     2a^-4:x'-3x-^ll:x-'2. 

84.  (a;^-3arV  +  a;/):(a;-32/):(a;  +  2/). 

85.  (2a^-\-10x^-7x^-Ux^-\-nx-2)  :  (x^-i-5x-2)  :  (a^-x+1). 

86.  (1  +  2  aj'^) :  (1  +  a;  4-  »^)  ;    (a;^"+  r ')  -  (a^"+  2/') .  each  to  four 

terms  of  the  quotient ;  write  the  complete  quotients. 
Make  two  di\isions,  the  first  to  ascending  powers  of  a;,  the 
other  to  descending  powers  of  x. 

87.  (a;^-2a;^4-3a.-3-4a;2_^5a._g).(^_2)^  the  quotient  :  (a;- 2), 

•  •• ;  write  the  last  quotient  and  the  numerical  remainders 
in  a  series  ;  use  sj^nthetic  division. 


§14.] 


EXAMPLES. 


73 


88-101.    Divide  as  shown  by  the  signs;   follow  the  processes 
given  in  the  notes  exemplified : 

§  12.      PROB.  4,  NOTE  5. 

88.  (aj*_2/^):  (ar^_2/2);      (4.x' -dif)  :  (2a^ -{-3f)  ; 

(a^»  —  /«)  :  (a;"  ±  y"") . 

89.  (a^--^^-):  (a  +  &);      (a^^  +  i  +  ^sn  +  i)  :  («  4.5), 

90.  (a«  +  a«6-'  +  ct^^>'  +  a'b'  +  b')  :(a*-^a'b  +  a'b'  +  ab'  +  6^) . 

91.  [(a^  +  a3).(ar^_a3^-|.  [ (ar^  +  aa;  +  a') •  (a?' -  aa;  +  a^) ] . 

92.  (f7T^'-c2)  :  (a  +  6  " 


c);     (a2_5_(.^).  («_5_^c) 


93.  {x-\-f-\-z'):  (x  +  y-\-z); 

94.  (a;""*  -  1)  :  {x""  -  1)  ;     (a;""* 


(a^-y-z)  :  (x-y-{-z). 
-1)  :  (a;"-l). 


.K-V 


§  12.      PROB.  4,  NOTE  6. 


55.  9/gr;(3/-\gr^±  2^2^2)2.  a+6  :  (a+6  ±  3 -aj+^z)';  by  mak- 
ing snitable  substitutions  in  Ex.  77.  Get  the  quotients 
as  far  as  the  cubes  of  u^v^  and  x-\-y,  then  write  the 
complete  quotients. 


m. 

a 

a^  +  2a- 

a^  +  2a3 

x-{-a'^ 

-^x^  +  a 

x-\-a^ 

+  b 

-\-4ab 

+  Ga'b 

+  Aa^b 

+  b 

+  2ab 

+  26^ 

+  Gab' 

4-6a2  62 

+  b' 

+  2b' 

+  4:ab' 

-hb' 

§  12.      PROB.  4,  NOTE  7. 

97. 

(a:2  +  2a;-13  +  2a;-i  +  ^'')  :  (a;  +  5  +  a;-i). 

98. 

(a-3  +  63  _f_  ^3  _  3ce5c)  :  (a  +  6  -f-c) ; 

99. 

la^(y-z)+f{z-x)-i-z^(x-y)^ 

:[^(2/-' 

z)  +  f{z~ 

x)  +  z'{x- 

-y)l- 

§  12.      PROB.  4,  NOTE  8. 

100.(l-.2a;4-.04a^-.008a^-f-..):(l  +  .la;-f--01a^+.001a^4----)- 
Carry  nothing  be3^ond  the  third  decimal  place,  nor  beyond  a^. 

101.  (ar^  +  lla^-102a;  +  181):(a;-3.213),  with  same  limita- 
tion as  in  Ex.  100. 


74  PEIMARY  OPERATIONS.  [II. 

102.  If  N  be  an}'  dividend;  Di,  Dg,  •••  any  divisors;  Qi,  %  the 

quotient  and  remainder  got  b}^  dividing  n  b}'  d^  ;  Q2,  R2 
the  quotient  and  remainder  got  b}-  dividing  Qi  by  D2,  •••, 
show  that  N,  =  Qi .  Di  +  Ri, 

=  (Q2D2  +  R2)  •  Di  +  Ri  =  Qo  •  D2D1  +  R2  •  Di  +  Ri, 
=  (Q3l>3  +  R3)  •  I>2l>l  +  R2  •  Dl  +  Rl 

=  QaDgDgDi  +  Rg  •  D2  Dj  4-  R2  •  Di  -f  Ri,  and  so  on, 

=  Qn  I>n  I>n-1  *  *  *  I>3l>2 1>1  +  R^  '  I>«-1  *  *  *  DgDi  H 

+  R2.DiH-Ri; 

and  if        Di  =  Dg  =  Dg  =  ••• ,  then  that 

N  =  Q„Di"  +  RnDi"-^  +  R„_iDi«--  +  ... 

4-R3i>i'  +  R2i>i  +  Ri. 

103.  By  the  method  of  Ex.  102  deye\o\yai^-\-8a^-\-24:af-\-32x-\-16 

to  powers  of  a;  +1  ;  of  a;  —1  ;  of  a;  +  3  ;  of  af  -j-  a;  +1  ; 
also  in  the  form    Aa;  -f  Ba;  (a;  +  1)  +  ca;  (a;  +  1)  (a;  +  2) 
+  Da;  (a;  +  1)  (a;  +  2)  (a;  +  3) ,  wherein  a,  b,  c,  d  are 
free  from  x. 

-.A.    -c^  3a;^  — 16a;2  +  24a;  — 1  „^      .. 

104.  Express       ; -— as  a  sum  of  fractions 

(a; -2)* 

whose  numerators  are  free  from  x. 
First  solution :  Develop  the  numerator  to  powers  of  a;  —  2 
[Ex.102],  viz.,  3(a;-2)3  +  2(a;-2)2-4(a;-2)+7; 
then, 
3a;3-16a;2_^24a;-l_3(a;-2)3      2(a?-2)^     4(a;-2)  7 


(a;-2)*  {x-2y  ^  {x-2y       {x-2y  '   (a;-2)^ 

3.2  4,7^^ 


x-2     {x-2y     {x-2y     {x-2f 
Second  solution :  Divide  both  numerator  and  denominator 
b}'   a;  —  2   three  times  in  succession  ;  then, 


3a;^-16ar^+24a;-1^3a^-10a;  +  4  7 

(aj-2)^  (a; -2)3       "*"(a;-2)* 

3a;-4  4  7 


(a; -2)2      (a; -2)3      (a; -2)* 
3.2  4         . 


x-2      {x-2y      (a; -2)3      (a? -2)* 


OF 
§  14.]  EXAMPLES.  75 

105.  Express      — ^ — ^,,^  as  a  sum  of  fractions  whose 

numerators  are  free  from  x. 

a^  4-  cc^  4-  cc  4- 1 

106.  Express      — — —     ^.,  as   a   sum   of  entire  terms, 

(a;-3)- 

and  of  fractions  whose  numerators  are  free  from  x. 

X* 4  a^  -I-  1 

107.  Express      — ,  ,     — -      as   a  sum  of  entire  terms 

x{x-{-l){x  +  2) 

and  of  fractions  whose  numerators  are  free  from  x,  and 
whose  denominators  are  ic,  x{x+l),  x{x-\-l){x-{-2); 
either  by  first  developing  the  numerator  as  to  a; +2, 
(a;  -h  1)  (a;  4-  2) ,  a;  (a;  +  1)  (a;  +  2) ,  or  by  dividing  both 
numerator  and  denominator  successively  by  aj  +  2, 
a;4-  1,  x. 

108.  Express       — as  a  sum  of  entire  terms 

^  a;(a;-l)(a;-2) 

and  of  fractions  whose  numerators  are  free  from  a;,  and 
whose  denominators  are  a?,  aj(a;— 1),  a;(a;— l)(a;— 2). 

§  13.     PROB.  5. 

109-112.    Reduce  to  lower  terms  the  fractions  : 

^-f-3^-f2.  a^— 3a; 4-2.  a^— 2a?— 15.  acx^-^(a4-bc)x—bd 
a^4-4a;4-3' a;2-4a;+3' a^4-2a;-35'  d'xF'-b'' 

a2_62  d'-b^  4x^-9 


109 
110 


a*-b^'       a^±2ab-hb^'      4a^±12a;4-9 


111  4a^-(3y-4;g)^  (4  a^  4-3  a;  4- 2)^- (2  a^  4- 3  a;  4- 4)^ 
*   {2x  +  3yy-16z''  (^3x'-\-x-iy-(x'-x-3y 

112  'TlSZHl.      P*  —  ^.      '^  —  ^.      a^" - y^"        o?n^y-2n 

§  13.      PROB.  6. 

113.  Reduce  to  equivalent  fractions,  with  the  common  numerator 

a*  —  6*,  the  fractions  : 
a-&.  a+b,  a^-b\  a''-^b\  o?-\-a'b+ab''+b\  o?-o?b+ab''-W 
a4-6'  a-&'  a'+b'''  a?-W'  a^-o?b+ab^-W'  o?+oj'b+ab''+b'' 


76  PEIMARY  OPERATIONS.  [11. 

114.  Reduce  to  equivalent  fractions  with  the  common' denomi- 


nator    Qi:^-\-a 

af  +  ab 

x  +  abc^ 

the  fractions : 

+  b 

-{■(tc 

+  c 

'  +  bc 

• 

a^-b 

x  +  bc           x^-a 

x  +  ac 

af-a 

X'{-ab 

—  c 

—  c 

-b 

x-\-a        '               x-^b 

x-\-c 

x—c                         x—b 

x  —  a 

x^  +  a 

a; -h a6  '         or -{-a 

x-\-ac  ' 

af  +  b 

x-\-bc 

-hb 

+  c 

+  c 

§  13.     PROS.  7. 
115.  Reduce  to  a  common  denominator  the  sets  of  fractions  : 
—    —   A'       ^  a;  ^  a^  ax 

xy  yz  zx'   1-ar*'  (l-a;)^'   a^  +  ax   o? -  ax   d'-x"'" 

a  3a         2ax         2  3  2a;  — 3  . 

a  —  x    a-haj    c?  —  o^''     x     2a;  —  1     4a;^  —  l' 

g^—  be  W  —  ca  (?  —  ab 

(a-|-6)(a  +  c)'    (6  +  c)(6  +  a)'    (T+^K^Tft)' 

§  13.      PROB.  8. 

Add  and  subtract,  as  shown  by  the  signs,  the  fractions : 

1 


116.   1+-1_+     1     ;    1 
1 +x      1—x 


1_.    1  — a;      1+a? 

1  +a;      1  —a;'   1  -\-x     1  —x 


-J-    a-\-b     a  —  b     a  -\-  b     a  —  b 


a  +  x     a  —  x'a  —  x     a-{-x'     a  +  6     a^  —  b^     a^  +  b^' 


118. 
119. 
120. 


1 


1 


1 


{a--b){a-c)      (b-c){b-a)      (c-a){c-b) 
a^  .  b^  .  (^ 


{a  —  b)'{a  —  c)      (6  — c).(6  — a)      {c  —  a)-(c-'b) 

__^!___^!_  of      .       x"^ 

a;«_l      a;"  +  i      a;"— 1      aj'^  +  l' 


§  14.]  EXAJVIPLES.  77 

§  13.     PROB.  9. 

122-124.  Multiply  and  divide,  as  shown  by  the  signs,  the  fol- 
lowing fractions.  Take  care  to  keep  every  fraction  in 
its  lowest  terms,  and  to  cancel  where  possible  : 

122.  ^^\^.^;     fi  +  lV  f._lV  fi_lY 
x  —  y       a  —  o       X  —  y       \       xj      \       xj      \      xl 

x*-h*  3^4- 6a;     a^^Tfs?     x"" -2hx^-\-V'x' 


x'-2hx-\-h^       x-b        a^-\-b^  x^-bx  +  b^ 

a^  —  a?  ^  a?  — a?  ^  a  —  x  ^  a?  —  ax-\-x^     a^-\-2ax 
a^  +  a^     o?-\-a?     a-\-x     a^-\-ax-\-x^     o?  —  2  ax -\- x^' 


125.  Show  that  a-.b:  c=^a'.b:  c  =  a-.b  xc,  [th.  3  cr.  8 


126.  Show  that  a:b:cid  —  a:b:c:  d 


=  a:b:cxd  =  a:bxcxd. 

127.  Show  that  ^.^S^  =  ^'^'^^"  =  ^-^''^' 
d  d'    d"     d'd'-.d"     d.d'-n" 


=  n  :  nTTd^ :  d  :  WT^=  n:  d:n' :  d' :dFT 


1 28.  Remove  the  bars  and  reduce  2a:  Sa^-  4  a^ :  6  a*  •  a^ :  a^  •  a. 

129.  Remove  the  brackets  and  reduce  to  lowest  terms : 

130.  Reduce  the  following  complex  fractions  to  simple  fractions, 
(a)  by  performing  the  operations  indicated, 

(6)  by  multiplying  both  numerator  and  denominator  by  a 
suitable  multiplier : 

x  —  y  a^—  b^        m^-\-mn-\-n^        p^  -{-q^ 

l-\-xy  a'-hb^  m^  +  n^  p^-\-(f^ 


..       x{x  —  y)  '       a  +6  '  m^  —  n^      '      p^  —  (f 


m-{-n  .  m  —  n 


1  +xy  a  —  b         m^—  mn  4-  n 

m-{-n 

m  —  n  '  m-\-n 
m  —  n  m  +  71 
m-^n     m  —  n 


1-1' 

1-1' 

1+1' 

1+aj 

1-x 

1+i 

X 

78  PRIMARY  OPERATIONS.  [II. 

131-136.   As  an  exercise  on  fractions,  prove  the  theorems  of 
proportion  [ths.  G-9]  and  their  corollaries. 

131.  If    -  =  -,  thenwmad  =  6c;    if  -  =  -,  then  wiU  ac  =  ft^. 

h     d  he 

132.  If  ad  =  be,  then  wiU  -  =  - ;      if  ac  =  b\  then  will   -  =  -. 

b     d  be 

133.  K    2  =  £,  then  will  ?^  =  ^,    »  =  ^,    £±^  =  £t^, 

b     d  a     c     c      d       a  c 

a±b  _c±d    a+_6_c-M. 
b  d    ^   a  —  b      c  — cZ' 

iQi    T^  a      c    a'     c'     a"      c" 
lo4.  li— =  -,   —  =  — ,    — = — ,    ..., 

b     d    b'     d''   b"     d" 
then  will  ^<^'<^"'"^cc'e"^:       oT^^e^^ 
bb'b"'"     dd'd"'"       6«     d« 

135.  K^  =  £  and  ^=''    then  will  ^:  ^  =  ^4,. 

b     d  b'     d'  b    b'     d   d' 

136.  K^  =  ^  =  ^=...,   then  will  ^  +  "  +  "  +  -=^  =  £  = 

b      d     f  6  +  CZ+/+...      b      d 

ha-\-7ce-\-le-\-'"  ^a      ha^  +  kc""  +  le"" -\-  •  - •  ^  a'' 
hb+kd  +  lf+'"      6'     /i6- -f-A-d'* +  ;/'•+...  ~  6«* 


and  conversely. 


§§  1-3. 

137.  State  the  converses  of  Axs.  1-7,  and  show  that  Ax.  1  is  its 

own  converse. 

138.  Show  that  of  an}'  simple  operation,  the  first  inverse  of  the 

first  inverse  is  the  original  operation ;  and  the  second 
inverse  of  the  second  inverse  of  the  second  inverse  is 
the  original  operation. 

139.  Show  that  if  a  simple  operation  be  commutative,  its  first 

and  second  inverse  are  alike  in  kind. 

140.  Exemplify  Exs.  138,  139  when  a;  =  operand,  A;  =  operator, 

u  =  result, 
and    u=x-{-k]   u=x—k;  u=2x-\-Sk;  u=kx:{k+x);   u=x^. 

Find  the  modulus  when  ^  =  ^^-±1;     when  u-  ^^^ 


x+k  2+k 


§  1.]  DEFmiTIONS.  79 


in.     MEASURES,  MULTIPLES,  AND  FACTORS. 

Many  of  the  properties  of  integers  are  shared  bj^  entire  literal 
expressions,  and  the  two  are  here  treated  together  as  entire 
numbers. 

§  1.     DEFINITIONS. 

When  the  complete  quotient  of  two  numbers  is  entire,  the 
I  divisor      .        ■  measure    ^  .,      i  dividend. 
'  dividend  '  multiple  '  divisor. 

E.g.,    0,  3,  G,  9  are  multiples  of  3  ;   0,  3^,  "7,  ^lOJ,  of  -3J ; 

and  3  is  a  measure  of  0,  3,  6,  9  ;    "3^,  of  0,  3^,  "7,  H^, 

So        a;  —  a  is  a  measure  of  or—  a^,  oc^—a^,  b{x  —  a),  but  not 

of  x-\-a', 

and  ar  —  a^,  a^  —  a^,b{x—a),  but  not  x-^a,  are  multiples 

of  a;  —  a. 

The  measures  and  multiples  of  a  numeral  depend  upon  its 

value  ;  of  a  literal  expression,  upon  its  form  ;  and  one  expression 

may  measure  another,  but  its  value  not  measure  the  value  of  that 

other,  or  the  reverse. 

.    E.g.,    if  a;  =  i  and  a==^, 

then         the  value  of  a;  —  a,  =  ^,  is  not  a  measure  of  the  value 

of  ay^  —  (J?,  =  |- ;  but  is  a  measure  of  the  value  of 

a;  +  a,  =  f . 

When  the  complete  quotient  is  entire  as  to  the  numerals,  or  as 

.  I  , .  y  ..        -1       ^1,     I  divisor     .        ,  measure    „  ,, 

to  any  letter  or  letters,  then  the  {  ^^^-^^^^^  is  a  \  j^j^i^ipi^  of  the 

\  A-  '  as  to  the  numerals,  or  as  to  the  same  letter  or  letters. 

'  divisor  ' 

E.g. ,    8  (ic  —  a)  (2/  —  &)  is  a  measure  of  2  m(a;^  —  a^)  {y  +  6) 
as  to  a,  m,  and  x, 
and  a  multiple  of  it  as  to  the  numerals  ; 

but  neither  measure  nor  multiple  as  to  ?/,  nor  as  to  b. 

So        \u~^  is  a  measure  of  f -y"^  as  to  the  numerals  and  w, 
and  a  multiple  as  to  v. 


80  MEASURES,   ISIULTtPLES,   AND  FACTOES.      [III.  th. 

Note.  The  words  "multiple"  and  "measure,"  as  here  used, 
are  an  extension  of  "multiple"  and  "part,"  as  used  in  I.  §8. 
When  the  quotient  is  an  integer  the  two  uses  are  identical,  but 
the}'  are  not  necessarily  identical  when  the  quotient  is  an  entire 
number ;  for,  though  an  integer  is  alwaj's  an  entire  number,  an 
entire  number  ma}'  or  may  not  be  an  integer. 

^  «—  ^  :X[:  °f  ^-  -  --  numbers  is  a  ^  --- 

of  each  of  them;   and  the  ^  '"'^''^f  <"""'"''™  "^'"fY^  is  that 

'  lowest  common  multiple 

-{  jnultinle  ^^^^^  gi^'^s,  for  quotient,  the  smallest  possible  nu- 
meral, or  the  literal  expression  of  lowest  possible  degree  and 
with  smallest  possible  coefficient. 

E.g.,  3  is  the  h.  c.  msr.  of  6,  9,  and  12,  but  not  of  6  and  12, 
and  18  is  the  1.  c.  mlt.  of  3,  6,  and  9,  but  not  of  3  and  9. 

So  a;  —  a  is  the  h.  c.  msr.  of  x^—  a^,  a^—  a^,  and  b(x  —  a), 
and  b{x^-a^)  is  the  1.  c.  mlt.  of  a^+a^,  a^—a^,  and  6(a;— a). 

Note.  Strictly  speaking,  two  or  more  numbers  have  two 
h.  c.  msrs.  and  two  1.  c.  mlts.,  opposites  of  each  other,  either  of 
which  may  be  used. 

E.g.,  3  and  —  3,  18  and  —  18,  x  —  a  and  a  —  x,  6(a;^  —  a"), 
and  6(a^  — a;^),  in  the  above  examples. 

An  entire  number  is  {  ^f''  when  {  ^  multUJle        ^f  2. 
'  odd  '  not  a  multiple 

E.g.,    —  6,  0,  2,  10 ab,  are  even  ;    ±  1,  3,  oa:^,  are  odd. 

-A-  -i  i  _       V   number  is  an  entire  number  that  has  ■{  ^^ 
•  composite  '  some 

entire  measure  besides  ±  itself  and  ±  1 . 

The  prime  factors  of  a  composite  number  are  the  primes  which, 
multiplied  together,  produce  it ;  and  to  factor  a  composite  num- 
ber is  to  find  all  its  prime  factors. 

E.g.,  GOOa'x"  -  600a'x'=  2^-3'5'-  -a^'x'-ia  +  x)'{a  -  x), 
twelve  prime  factors. 

Entire  numbers  are  prime  to  each  other  when  they  have  no 
enth-e  common  measure  except  ±1. 

E.g.,  9,  10,  a^,  x^,  7?/,  a^  —  y^,  are  all  composite,  but  prime 
to  each  other. 


1.  §  3.]  AXIOMS  ;    MEASURES  AND  MULTIPLES.  81 

§  2.     AXIOMS. 

1 .  Every  number  is  both  a  measure  and  a  multiple  of  itself, 
and  of  its  opposite. 

2.  ±  1  is  a  common  measure  of  all  entire  numbers,  and  a  com- 
mon multiple  of  their  reciprocals. 

3.  0  is  a  common  multiple  of  all  other  numbers. 

Ay       .  I  measure  of  a  numeral  not  0,   .      ,  ,       ,        .  small 
^^^y  '  multiple,  not  0,  of  a  numeral,  '  large 

as  that  numeral. 

5.  Every  multiple,  not  0,  of  a  number  contains  all  the  letters 
of  that  number,  and  to  at  least  as  high  a  degree  as  the  number 
itself. 

6.  All  prime  numbers  are  prime  to  each  other ;  but  not  all 
numbers  prime  to  each  other  are  primes. 

7.  ±  1  is  a  prime,  and  is  prime  to  all  entire  numbers. 

8.  An  entire  function  of  entire  numbers  is  an  entire  number. 

§  3.    MEASURES  AND  MULTIPLES. 

rj^  t       A       \  Tneasure     -  t,      •       ;  measure    j, 

Theok.  1.    Any  \  ^^^^^^.^^^  of  a  number  isa-{  ^^^^^.^^  of  any 

.  multiple  ^^  ^j^^^  ,^^^j^^_ 

»  measure   -' 

-r   ^     ,  n  -,  I  Mamsr.    «      .t       •    i  Mamsr.  ofN. 

Let  A  beiiny  number,  and  ^  ^  ^  ^^^  of  a,  then  is  ^  ^  ^^  ^^^  ^^^^ 

For     *.•  the  quotients  a  :  m  and  n  :  a  are  entire,  [hyp. 

.-.  A  :  M  X  N  :  A,  =  N  :  M,  is  entire.  q.e.d.    [ax,  8 


Note.    Th.  1  may  also  be  stated  thus : 

,       .        I  measui 
finer  is  a<        ,.. 


A       I  measure  of  a  measure  ^«      „ ,  ^„  .    „  i  measure     « 

^°y^  multiple  of  a  multiple  °^  "  ""'"'"^'^  '^  ''^  multipk  ""^ 

that  number. 

r^        ^        A       \  tneasure     -  t,       •        ;  measure  ^r.  .,„ 

Cor.  1.     Any{  ^^^^^^^  of  a  number  zs  a  {  ^^j^^^j^  of  Us 

opposite. 

CoR.  2.     An  even  number  cannot  measure  an  odd  number; 

,  measure    -       ,  odd       _r^^  -^  i  odd. 
I.Q.,  any  <        ,,.  ,    of  an  <  number  is  <  ^^.^^ 

'      -^  '  multiTole   -^        »  even  '  even. 


82  MEASUKES,  IVIULTIPLES,   AND  FACTOES.     [III.  tlis. 

Theor.  2.    A  common  measure  of  two  or  more  numbers  is 
a  measure  of  their  sum. 

Let  A,  B,  •••  be  any  numbers  and  m  a  common  measure  o^ 

them,  then  is  m  a  measure  of  a  +  b  -^ . 

For     •  .*  A  =  a  •  M,  B  =  6 .  M,  •  •  • ,  wherein  a,  &,  •  •  •  are  entire,  [hj^p. 

.*.  a  +  bH =a-M4-6.MH [II.ax.2 

=  (a4-&  +  --)-M;  [II.th.4 

and    •••  a  +  6 +  •••  is  entire,  [ax.8 

.•.(a  +  b  +  ...):m,    =a.f6  +  ...,  [I.  §9df. 

is  an  entu*e  number.  q.  e.  d. 

Cor.  1 .    A  common  measure  of  two  numbers  is  a  measure  of 

their  difference. 

Cor.  2.    If  a  number  measure  the  sum  of  two  or  more  num- 
bers^ and  measure  all  but  one  of  them,  it  measures  that  one  also. 
Cor.  3.    A  common  measure  of  two  or  more  numbers  is  a 
measure  of  the  sum  of  any  multiples  of  them. 

Cor.  4.    A  common  measure  of  two  numbers  is  a  measure  of 
the  difference  of  any  multiples  of  them. 

Theor.  3.    If  a  simple  monomial  measure  a  polyjiomial  in  its 
simplest  foron,  the  monomial  is  a  common  measure  of  all  its  terms. 

Let  a  +  bH be  a  polj'nomial  whose  terms  are  simple  and 

unlike,  and  let  m,  a  monomial,  be  a  measure  of  it ;  then  is  m  a 

common  measure  of  the  separate  terms  a,  b,  •••. 

For     *.*  A,  B,  •••  are  unlike,  [byp. 

.*.  the  quotients  a  :  m,    b  :  m,  •••  are  unlike, 
and  their  sum  cannot  be  reduced  ; 

.'.if  either  of  them  were  fractional, 
then         their  sum,  the  quotient  (a  +  b  H )  :  m,  would  be  frac- 
tional ; 

^ut     *.- (a  +  B  H )  :  M  is  entire  and  not  fractional,  [hyp. 

.*.  A  :  M,  B  :  M,  •••  are  not  fractional,  but  all  entire.  Q.  e.  d. 
Note.   Ths.  1,  2,  3,  and  their  corollaries,  may  be  extended  by 

...       I  "measure       ,  ,  , ,  ,  ^,       „       ,  ,  "measure 

'^^''S  ■!  u  multiple  as  to  any  letter  or  letters,"  and  ^  ^^^^^^^^i^ 

as  to  the  same  letter  or  letters,"  instead  of  simply  ■{  ii^^^f."^*^',, 


2-4.  §4.]    PEIjNIE  AND  COMPOSITE  NTBIBERS — FACTORS.      83 

§4.     PRIME  AND  COMPOSITE  NUMBEES. -FACTORS. 

Theor.  4.  If  a  prime  number  measure  the  product  of  two  or 
more  entire  numbers^  it  measures  at  least  one  of  them. 

Let  A,  B,  •••be  any  entire  numbers,  and  let  p,  a  prime,  measure 
their  product ;  then  will  p  measure  either  a,  or  b,  or  some  other 
one  of  them. 

(a)  A  and  b  two  numerals,  p  a  numeral. 
For,  if  not,  divide  a  and  b  by  p,  and  let  Q,  q'  =  the  quotients, 
and  R,  r'  =  the  remainders,  all  integers  ; 

then    •••   A  =PQ-f-R    and   b  =  pq'  +  r',  [I.  §  9  df. 

.-.  ab  =  p^qq'  +  pqr'  +  pq'r  +  rr',  [II.  ax. 4 

i.e.,  ab  =  a  multiple  of  p,  +  rr',  [th.  2 

.-.  p,  a  measure  of  ab,  also  measures  rr'.  [th.  2  or.  2 

Divide  p  by  r,  and  let  Qi,  Ri  =  quotient  and  remainder,  both 

integers ; 

divide      p  by  Rj,  and  let  Qo,  R2  =  quotient  and  remainder,  both 

integers,  and  so  on  ; 
then   *.•  R,  Ri,  R2,  •••  are  all  integers,  and  successively  smaller 
and  smaller, 
.*.  one  of  them,  say  r^,  is  0  ; 
and    •.*  p,  a  prime,  when  divided  by  R;^.!,  the  next  preceding 
remainder,  gives  r^,  =  0,  for  remainder, 
.-.  R*.i  =  l. 
But-.*  Ri    =p  — QiR,   R2  =  p  — Q2R1,  •••  5  [above 

.-.  Rir'  =  pr'  — QiRr',   R2r'  =  pr'  — Q2RiR'j  ••• ;    [11.  ax. 4 
and    *.•  p  measures  rr',  [above 

.*.  p  measures  Rjr',    =  pr'  —  QiRr',  [th.  2  cr.  4 

.'.  p  measures  RgR',    =  pr'  —  Q2RiR'j    and  so  on  ; 
.*.  p  measures  r^.ir',  =  r'  ;  [above 

i.e.,         p  measures  a  numeral  <  itself,  which  is  absurd,     [ax.  4 

.*.  the  supposition  that  p  measures  neither  a  nor  b  fails, 
and  it  is  only  left  that  p  measures  one  of  them.  Q.  e.  d. 

(6)  A  and  b  two  numerals,  p  a  literal  expression. 
This  case  cannot  occur,  since  the  numerical  product  a  •  b  can- 
not be  measured  by  an  entire  literal  expression.  [ax.  5 


84  MEASURES,  MULTIPLES,  AND  FACTOKS.         [III.  th. 

(c)  A,  or  B,  or  both  ofthem^  literal  expressions,  p  a  numeral. 
For,  let  a;,  y,  •••  be  the  letters  involved  in  a,  or  in  b,  so  that 

A,  or  B,  or  both  of  them,  are  functions  of  a;,  ?/,  •••  ; 
then         if  p  does  not  measure  a  nor  b,  p  does  not  measure  all 
the  terms  of  a,  nor  of  b.  [th.  2 

Of  those  terms  of  ^  ^  which  are  not  measured  by  p,  let  -{  ?  be 

the  numerical  coefficient  of  that  one  whose  degree 
as  to  X  is  highest, 
or  if  there  be  two  or  more  such  terms,  then  of  that  one  of 

them  whose  degree  as  to  y  also  is  highest,  and 
so  on ; 
then   •.*  a  •  6  is  the  coefficient  of  a  term,  t,  of  a  •  b,  which  has  a 
higher  degree  as  to  its  own  letters  x,  ?/,  •••  than 
the  degi-ee  of  an}^  other  term  as  to  those  letters, 
and  is  like  no  other  term  ; 
.'.  T  combines  with  no  other  term,  and  remains  unchanged 
when  the  polj'nomial  is  in  its  simplest  form. 
But  *.•  p  measures  neither  a  nor  6,  [bjp. 

.*.  p  does  not  measure  their  product  a*  6,  [(a) 

.*.  not  the  term  t  of  a  •  b, 

.-.  not  A-B,  [th.3 

which  is  contrary  to  the  hypothesis  of  the  theorem  ; 
.*.  the  supposition  that  p  measures  neither  a  nor  b  fails, 
and  it  is  only  left  that  p  measures  one  of  them.  q.  e.  d. 

(d)  A,  or  B,  or  both  of  them,  literal,  p  literal. 
For,  if  not,  let  a;  be  a  letter  found  in  p ; 

then         X  is  found  also  in  the  product  a  •  b,  [ax.  5 

and     .*.  in  either  a,  or  b,  or  both  of  them. 

Arrange  a,  b,  and  p,  by  descending  powers  of  a;,  divide  a  and 
b  by  p,  and  let  q,  q'  =  the  quotients,  and  r,  r'  =  the  remainders, 
all  entire  as  to  a; ; 
then   -.-.A    =PQ  +  R   and  b  =  pq'  +  r',  [I.  §9,df. 

.-.   AB  =  P^QQ'  +  PQR'-f-PQ'R  +  RR',  [II.  ax.  4 

i.e.,  ab  =  a  multiple  of  p  as  to  x,  +  hr'?         [th.  2,  th.  3  nt. 

.*.  p,  a  measure  of  ab,  also  measures  rr'  as  to  x.  [th.  2  cr.  2 


4.  §4.]    PRBIE  AND  COMPOSITE  NUMBEKS.  — FACTORS.  85 

Divide  p  b}"  r,  and  let  Qi,  Ri  =  quotient  and  remainder,  both 
entire  as  to  ic ; 
divide      p  by  Ri,  and  let  Qs,  R2  =  quotient  and  remainder,  both 

entire  as  to  a;,  and  so  on  ; 
then   •••  p,  being  prime,  has  not  x  in  every  term,  [th.  2 

and     •••  R,  Rj,  Ro,  •••  are  entire  as  to  x,  and  of  successively  lower 
and  lower  degree, 
.*.  one  of  them,  say  Rj_i,  is  free  from  x. 
But*.-  Rj     =p    —  QiR,      R2     =P    —  Q2R11  •••?  [above 

.-.  Rir'  =  pr' —  QiRr',   R2r'  =  pr'  — Q2R1R',  •••,     [II.  ax. 4 
and    *.•  p  measures  rr'  as  to  x,  [above 


p  measures  Rjr',  =  pr'  —  QiRr',  as  to  x 


.'.  p  measures  R2R',  =  pr'  —  Q2RiR\  as  to  x,  and  so  on  ; 

.'.  p  measures  r^-iR',  as  to  a; ; 

i.e.,  p  measures  an  expression  of  lower  degree  as  to  x  than 

p  itself,  which  is  absurd  ;  [ax.  5 

.*.  the  supposition  that  p  measures  neither  a  nor  b  fails, 

and  it  is  only  left  that  p  measures  one  of  them.     q.  e.  d. 

(e)    Three  or  more  factors,  a,  b,  c,  •••  l. 

For,  if  p  measures  the  product  a  •  b  •  c  •••  l, 

then         p  measures  either  a  or  the  product  B'C  •••  l,       [above 

if  p  measures  the  product  b  •  c  •••  l, 

then         p  measures  either  b  or  the  product  c  •  •  •  l,  [above 

and  so  on ; 

.-.  p  measures  either  a,  or  b,  or  c,  or  ••• ,  or  l  ; 

i.e.,         p  measures  one  of  them.  q.  e.  d. 

Cor.  1 .     If  a  prime  measure  the  product  of  two  numbers,  and 

he  prime  to  one  of  them,  it  measures  the  other. 

Cor.  2.     If  there  he  two  or  more  entire  numhers,  and  if  p,  a 

prime,  measure  neither  of  them,  it  does  not  measure  their  product ; 

and  if  not  their  product,  then  neither  of  them.     In  particular,  a 

prime  cannot  measure  a  product  of  other  primes. 

n        o        A        -j*^^«^*        J  some  even  .    .  even. 
Cor.  3.     A  product  of  entire  factors  ■{    ■,.     -,-,       is  ^  ^^^ 

Cor.  4.  If  a  prime  measure  a  positive  integral  power  of  an 
entire  numher,  it  measures  that  number;  and  if  the  number,  then 
the  power. 


86  MEASURES,  INIULTIPLES,  AXD  FACTORS.         [III.  th. 

Theor.  5.  A  composite  number  can  be  resolved  into  one,  and 
but  one,  set  of  prime  factors. 

(a)  Into  one  set. 

For,  let  N  be  any  composite  number,  m  an  entire  measure  of 
N,  and  Q  the  quotient, 
then   •••  N  =  M.Q,  [I.  §9df. 

.-.  if  M  and  Q  be  primes,  N  is  resolved  as  required. 
But  if  either,  or  both  of  them,  be  composite, 

then         they  also  may  be  resolved,  and  so  on. 
Finally,  when  all  the  prime  factors,  a,  b,  •••,  are  found,  if  a 
occur  a  times,  b  b  times,  •••, 

then  N  =  A*'-B*--'.  Q.E.D. 

(6)  Into  but  one  set. 

For,  if  possible,  let  n  =  a*-b'-",  also  =G^-n*---,  wherein 
A,  B,  •••  are  unequal  primes,  and  so  are  g,  h,  ••• ,  and  g*,  h*,  ••• 
are  not  wholly  the  same  as  a",  b',  ••• ; 

then    *.•  some  prime  p  occurs  p  times  in  one  set,  and  not  p  times 

in  the  other  set, 

. •.  of  the  equal  quotients  a"  •  b*  •  •  • :  ?*»  and  g*'  •  n'*  •  •  •  :  p^  one 

is  entire  and  the  other  fractional,  which  is  absurd ; 

.*.  the  supposition  fails  that  n  can  be  resolved  into  two 

different  sets  of  prime  factors.  q.  e.  d. 

ENTIRE  NUMBERS  PREME   TO   EACH   OTHER. 

CoR.  1.  If  tico  entire  numbers  have  no  common  prime  factor^ 
they  are  prime  to  each  other. 

Cor.  2.  If  there  be  two  sets  of  entire  numbers,  such  that  each 
number  of  the  first  set  is  prime  to  each  number  of  the  second  set, 
then  is  the  product  of  the  first  set  prime  to  the  product  of  the  second 
set;  and  conversely. 

CoR.  3.  If  two  entire  numbers  be  prime  to  each  other,  so  are 
any  positive  integral  powers  of  them  ;  and  conversely. 

CoR.  4.  If  there  be  two  entire  numbers  prime  to  each  other, 
any  common  multiple  of  them  is  a  multiple  of  their  product. 

For,  let  the  products  a"  •  b*  •  •  • ,  g''  •  h*  •  •  • ,  be  any  numbers  prime 
to  each  other,  and  let  m  be  a  common  multiple  of  them ; 


5.  §4.]       PEIME  AND  COIklPOSITE  NUMBERS.— FACTOES.      87 

then    •••  among  the  prime  factors  of  m,  a  occurs  a  tunes,  b  b 

times,  •••,  G  g  times,  h  h  times,  •••, 
and     •••  the  primes  a,  b,  •••,  g,  h,  •••  are  all  different,        [hyp. 

.-.    M=Q  X  A^-B*---  X  G^-H*---, 

wherein   q  is  some  entire  number,  perhaps  1.     q.e.d.  [II.  th.  3 
So  if  there  be  three  or  more  entire  numbers  prime  to  each  other. 

COMMON  MEASURES  AND  MULTIPLES. 

h  c  Ttisr 
Cor.  5.     TJie  ^  , '   *    ^  '  of  two  or  more  entire  numbers  is  the 
'  I.  c.  mlt.     '' 

product  of  their  different  prime  factors^  each  factor  having  the 

-l   ^f  ^  .    .  exponent  which  it  has  in  any  of  the  numbers, 
greazesz 

Cor.  6.    If  there  be  two  or  more  sets  of  entire  numbers^  the 

h  c  msr 
of  all  the  given  numbers.     In  particular^  the  ■{  i'    'j.  '  of  three 

or  more  numbers  is  the  ■{  .'    '    ]*  '  of  any  one  of  them  and  the 

,  h.c.msr.     /..t      ^i 
-s  7         7*     of  the  others. 
'  I.  c.  mlt.     -' 

Cor.  7.    If  each  of  two  or  more  entire  numbers  be  multiplied 

h  c  Tnsr 
{or  divided)  by  any  same  number,  their  -j  ,  *  '    j.  '  is  multiplied 

(or  divided)  by  that  number. 
CoR.  8.     TJie  h.  c.  msr.  of  two  entire  numbers  is  not  changed 

when  either  of  them  is  ^  ^"[^f^'"^  b>/  an  entire  J,  "^^^^^^  prime 

to  the  other. 

For    •.•  the  prime  {  """^'='^  multiplied  into   ^  ;  ^  f^^. 

^  '  measure  stricken  out  of 

tor  of  the  other, 
.-.  it  -{  ^^  °?|      t  b   ^  factor  of  their  h.  c.  msr.  Q.  e.  d. 

Cor.  9.  TJie  product  of  two  entire  numbers  equals  the  prod- 
uct of  their  h.  c.  msr.  and  I.  c.  mlt. 

For,  let  N,  =A«.B^..-,  and  n',  =a'''.b^'  •••,  be  any  two 
numbers,  wherein  a,  b,  •••,  are  primes,  and  the  exponents  a,b.  •••, 
a',  6',  •  •  • ,  are  integers, 


88  MEASURES,  MULTIPLES,  AND  FACTORS.        [III.  th. 

then   •••  of  the  exponents  a,  a',  the  ■{   ^^^  .      is  the  exponent  of 
»  ^r.  fi.«  J  ^'  c.  msr.        greater 
A  in  the  <  1  ,, 

'  1.  c.  mlt. 

.*.  a  +  a'  is  the  exponent  of  the  factor  a  in  the  product  of 

the  h.  c.  msr,  by  the  1.  c.  mlt.  [II.  th.  3  cr.  10 

So        6  +  6'  is  the  exponent  of  b  in  that  product,    and  so  on  ; 

.*.  the  product   h.  c.  msr.  X  1.  c.  mlt.  =  a"+*'«b*  +  ^'  •••. 
But-.-  n-n'  =  a»+«'.b'  +  ''-..,  [II.  th.3 

.-.  N  •  n' =  h.  c.  msr.  X  1.  c.  mlt.  q.e.d.   [II.  ax.  1 

Cor.  10.    Every  common  \       ,..  ,    of  two  or  more  numbers 

t  measure    ^ .,   .    ,  h.c.  msr. 
'"^■^  multiple  "-f "'''"- <l.c.r>ilt. 

APPLICATION  TO  FRACTIONS. 

CoR.  11.  If  the  terms  of  a  simple  fraction  be  prime  to  each 
other ^  the  fraction  cannot  be  reduced  to  an  equivalent  simple  frac- 
tion in  lower  terms. 

A*  •  B*  •  •  • 

For,  let  be  a  fraction,  wherein  a,  b,  •••,  g,  h,  •••  ai*e 

G^-II*---  p 

all  different  primes,  and  let  -  be  any  equivalent  simple  fraction  ; 

Si 

then   -.- — =-,  [hyp. 

.-.  a".b'---  X  q  =  g^-h*  •..  X  P,  [II. ax.4 

whose  two  members,  being  the  same  number,  can  be 

factored  in  only  one  wa}' ;  [th. 

.*.  among  the  factors  of -j      are  ^     »'    »'      ' 

.-.  ^^  is  a  multiple  of  ^^;;^I;;:^ 

P  A*  •  B^  •  •  • 

and  -  is  not  in  lower  terms  than q.e.d. 

Q  G^-H*.-- 

CoR.  12.  If  a  fraction  be  in  its  lowest  terms^  so  is  every  integral 
power  of  it;  and  conversely. 

CoR.  13.  A  fraction  can  be  resolved  into  but  one  set  of  factors 
and  divisors,  a",  b^,  •••,  wherein  a,  b,  •••  are  primes,  all  different, 
and  a,  b,  •••  are  integers,  some  of  them  negative. 

Note.  By  aid  of  Cor.  13,  Cors.  5-10  are  extended  and  applied 
to  fractions  as  well  as  entire  numbers. 


5.  §4.]        PEtNIE  AND  COMPOSITE  NUI^IBERS.— FACTORS.       89 

Cor.  14.    The  ■{  j\\  u'  of  two  fractions  is  a  fraction  wJiose 
numerator  is  the  •{  -. '   '..  '  of  their  numerators,  and  whose  de- 
nominator is  the  ^  t'    '        'of  their  denominators. 
»  h.  c.  msr.   -' 

For,  let  -,  =  a"-b^««',  — ,  =  a^'-b^'  •••,  be  an}'  two  fractions, 
D  d' 

wherein  a,  b,  •••  are  primes,  and  the  exponents  a,  6,  •••,  a\  6',  ••♦, 
are  integers,  some  of  them  negative  or  zero ; 

and  let    \  ^^'  ■^''  "'  be  the  {  ^^  ^    ^  exponents  in  the  pairs  of 

exponents  a,a\   6,6',    ••• ; 

.X.  Au     )  h.  c.  msr.  ,  ^i    i.  •     •     i  both 

then   •.•  the  \  ,  ,.      has  every  measure  that  is  m  \    .  , 

of  the  fractions,  and  has  no  others,  [§  1  dfs. 

,,      1  h. c. msr.    ^  n  n'   .    .,  ,     .  ,  a"i«b''i  •••, 

•■•  ^^^  -i  1.  c.  mlt.   °f  5'  D-'  '"  ^^^  P™'^"''*  ^  A'^.B'. ...; 
wherein   those  factors  which  have  negative  exponents  make  up 
the  denominator  of  the  \  ^•^•"^^'^-  sought,  q.e.d. 
So  for  three  or  more  fractions. 

PRIME  AND  COMPOSITE  MEASURES. 

Cor.  15.     The  entire  number  a"-b*  •••  has  (a+  l)-(b  +  1)  ••• 
different  entire  measures,  prime  and  composite  {and  their  oppo- 
sites),  ivhose  sum  is  [(a«+^-1)  :  (a-1)]  •  [(b^+^-I)  :  (b  — 1)]  .... 
For     •••A"  has  (a+l)  measures,  a*,  a''-\  A"-^  ...,  a^  a^  1, 
and     •.•  B^  has  (6+1)  measures,  and  so  on, 
and     *.-  the  several  products  got  b}-  multipljing  the  (a  +  1) 
measures  in  turn  by  the  (6+1)  measures,  and  so 
on,  are  all  different  one  from  another, 
and     *.•  there  are  (a  +  l).(6  +1)  .••  of  these  products,  all  told, 
.*.  there  are  (a+l).(6+l)  .••  different  measures,  q.e.d. 
And   •.*  the  sum  of  the  measures  is  the  sum  of  all  the  different 
products  of  the  measures  a",  a""^  .«.,  a\  1,  b}^  the 
measures  b*,  B^~^  •.«,  b\  1,  bj-  ••., 

.-.  the  sum  =  (a"  H f-l)-(B*H h  1)  •••        [IL  th.5 

=  [(A-+^-l):(A-l)].[(B^+^-l):(B-l)]....    [II.  6 


90  IMEASURES,  MULTIPLES,  AND  FACTORS.         [III.  pr. 

§  5.   PROCESS  OF  FINDING  THE  HIGHEST  COMMON  MEASURE. 

PrOB.  1.  To  FIND  THE  HIGHEST  COMMON  MEASURE  OF  TWO 
OR   MORE    NUMBERS. 

(a)    The  prime  factors  of  all  the  numbers  known : 
Multiply  together  all  the  different  prime  factors^  each  with  the  least 
exponent  it  has  in  any  one  of  the  numbers,     [th.  5  cr.  5,  cr.  13  nt. 
E.g..,   of  9a^6*c,    Sabred,   and  15 a6^c^— 12 aV,  the  common 

prime  factors  are  3,  a,  6-;  the  h.  c.  msr.  is  3  •  a  •  b^. 
So        ofixy~^,  ^^y-,   '^^y~'  {x  +  y)  theh.  c.  msr.  is  Ja;?/"". 
{b)    The  prime  factors  not  known;  two  entire  numbers: 
Divide  the  higher  number  (the  larger  if  a  numeral,  and  that  of 
higher  degree  if  literal)  by  the  lower ;  the  divisor  by  the  remainder^ 
if  any ;  that  divisor  by  the  second  remainder,  and  so  on^  till  noth- 
ing remains. 

At  'nlPo.^re  ^  suppress  from  any  divisor,   ^^^.g^.^y.  factor  that 
At  pleasure,  ^  ^^^^^^^^^^  ^-^^^  any  dividend,^""'^  ^""^'^ ^^"^^^^  ^"^^ 

.    .,     I  dividend  ■,. 

IS  prime  to  the  ■{  ^i^^^q^    corresponding. 

At  pleasure^  suppress  from  any  divisor  and  the  corresponding 
dividend,  any  common  measure  of  them;  but  reserve  it  as  a  factor 
of  the  final  residt. 

The  last  divisor,  as  above,  multiplied  by  the  reserved  factors, 
if  any,  is  the  h.  c.  msr.  sought. 

Let  A  and  b  be  an}^  two  numbers,  a  the  higher,  Q  the  quotient 
of  A  b}'  B  ;  Ri,  R2,  Rg,  •••,  R„_i,  ««?  the  successive  remainders, 
whereof  r„  is  a  measure  of  r«_i  ;   then  is  r«  the  h.  c.  msr.  sought. 

1.    If  no  factors  be  introduced  or  suppressed. 
For     •••  Ri  =  A-QB,  [I.  §9df. 

.*.  whatever  common  measures  a  and  b  have,  the  same 
measures  has  Ri  ;  [th.  2  cr.  4 

but       •.•    A  =  Ri  +  QB, 

.'.  whatever  common  measures  b  and  Ri  have,  the  same 
measures  has  a,  [th.  2  cr.  3 

.*.  whatever  common  measures  b  and  Ri  have,  the  same 
common  measures,  and  no  others,  have  a  and  b  : 


1.  §5.]      FINDING  THE  HIGHEST  COMMON  MEASURE.  91 

SO  whatever  common  measures  %  and  R2  have,  the  same 

and  no  others  have  b  and  %, 
.-.  the  same  and  no  others  have  a  and  b,    and  so  on  ; 
so  whatever  common  measures  r„_i  and  r„  have,  the  same 

and  no  others  have  r„_2  and  r„_i,  the  same  and  no 
others  have  R^.g  and  r„_2,    and  so  on, 
the  same  and  no  others  have  a  and  b  ; 
but     •••  R„  is  the  h.  c.  msr.  of  r„_i  and  r„,  [h^'P- 

.  R„  is  the  h.  c.  msr.  of  a  and  b.  q.  e.  d. 

If  factors  not  common  he  introduced  or  suppressed. 
For     • .  •  the  h.  c.  msr.  of  the  given  polynomials  is  that  of  any  two 
successive  remainders  of  the  series,  [1 

and     •.*  the  h.  c.  msr.  of  these  remainders  is  not  changed  when 
either  of  them  is  modified  by  the  introduction  or 
suppression  of  a  factor  prime  to  the  other  ;  [th.  5  cr.  8 
.-.  the  h.  c.  msr.  of  these  two  modified  remainders  is  the 
h.  c.msr.  sought. 
So         for  an}^  two  remainders  subsequent  thereto. 
So         for  the  modified  r„_i  and  r„.  q.  e.  d. 

3.   If  a  common  factor  he  suppressed  and  reserved. 
For     • .  •  the  h .  c.  msr.  of  the  given  poljmomials  is  that  of  any  two 
successive  remainders  of  the  series,  [1 

and  • .  •  when  both  of  these  remainders  are  modified  b}'  the  sup- 
pression of  a  factor  common  to  them,  their  h.  c.  msr. 
is  divided  by  the  same  factor ;  [th.  5  cr.  7 

.  • .  the  product  of  the  h.  c.  msr.  of  these  two  modified  remain- 
ders by  the  suppressed  factor  is  the  h.  c.  msr.  sought. 


So         for  an}'  two  remainders  subsequent  thereto. 
So        for  the  modified  r„_i  and  r„. 

Q.E.D. 

E.g.^     to  find  the  h.  c.  msr.  of  a?-\-  x  — 

12   and  fl^-10a;  +  21. 

a^  +  x-12 

a^_10a;  +  21 
a.2-|.     a; -12 

1       or 

1     1  -12 

1   -10     21 
1       1  -12 

\-Ux-hS3(-n 
a^—Sx                      X—    S\x+4: 

1  -3 

Ll)-ll     33 
1     -3 

4a;-12 

4  -12 

d          a;  —  3  is  t 

he  h.  c.  msr.  so 

ught. 

Q.E.F. 

92  MEASURES,  MULTIPLES,  AND  FACTOKS.       [III.  pr. 

So        to  find  the  h.  c.  msr.  of  4  aa?  +  4  arc  —  48  a  and 
4aa^  — 40«a;  +  84a: 
*.•  4a  is  a  common  factor, 
and     *.•  of  the  remaining  factors,  cr' -fa;— 12  and  a^— 10  a; +  21, 
the  h.  c.  msr.  is  a  —  3,  [above 

•••  4  a  (a;— 3),  =4aa;— 12  a,  is  the  h.  c.  msr.  sought,  q.e.f. 

(c)  The  prime  factors  not  known  ;  three  or  more  entire  numbers : 
Find  the  h.  c.  msr.  of  any  two  of  them  (preferably  the  two 

lowest),  then  the  h.  c.  msr.  of  this  measure  and  the  next  number, 

and  so  on  till  all  are  used;   the  h.  c.  msr.  last  found  is  the 

h.  c.  msr.  sought.  [th.  5  cr.  6 

E.g.,   to  find  the  h.  c.msr.  of  0^+ a;  — 12,  a^  —  10a; +  21,  and 

iBS-_6ar^-19a;  +  84: 

•.•  of  a;^+a;— 12  and  ar'— 10a; +  21  the  h. c.msr.  is  a;— 3, 

and     •.•  a;  — 3  measures  a;'^  — 6a;^— 19a;  +  84, 

.•.  a;  — 3  is  the  h. c.msr.  sought.  q.e.f. 

(d)  Some  or  all  of  the  numbers  fractions : 

Divide  the  h.  c.  msr.  of  the  entire  numbers  and  the  numerators 
by  the  I.  c.  vrdt.  of  the  denominators.  [th.  5  cr.  14,  pr.  2 

E.g. ,    to  find  the  h.  c.  msr.  of  — — and — — : 

x  —  6  x  +  5 

'.•  the  h. c.msr.  of  the  numerators  is  a;  — 3,  [above 

and     •.*  the  1.  emit,  of  the  denominators  is  a;^— 25,    [inspection 

Q. 3 

.*.  — is  the  h. c.msr.  sought.  q.e.f. 

ar  — 25 

Note  1.  In  the  process  of  case  (b)  each  of  the  remainders 
Ri,  R2,  •••  is  the  sum  of  a  multiple  of  the  first  number  and  a 
multiple  of  the  second  number. 

Note  2.  The  arrangement  of  terms  ma}'  be  as  to  the  ascending 
powers  of  some  letter,  or  as  to  the  descending  powers,  at  pleasure. 

E.g.,    2a^+lla^+20a; +21     and    a^-x-S, 
or  21  +20a; +lla;2^2a;3  and    6 +a;  — a;^. 

That  arrangement  is  commonl}'  best  which  makes  the  trial 
divisor  smallest ;  and  at  an}^  step  of  the  work  the  highest  or  lowest 
term  of  the  divisor  may  be  used  as  trial  divisor  at  pleasure. 


1.  §  5.]      FINDING  THE  HIGHEST  COIOION  MEASURE. 


93 


The  work  is  often  shortened  by  using  detached  coefficients, 
and  sometimes  b}^  synthetic  division.  It  is  also  shortened  by 
arrangement  in  columns  and  b}'  not  writing  down  quotients  and 
products,  but  onl}-  remainders. 

E.g.^    to  find  the  h.c.msr.  of  2aj^+a;^— 4a;— 3  and  2x^—^x-\-b: 


and 
So 


or 


So 


and 


4) 


1 

5 

-4 
3 

-3 

2  -5 

6 

—  7 

6- 

-15 

9 

8- 

-12 

2 

-3 

2  -3 

3  or  2 


1  ■ 

6- 

-4 

—  7 

-3 

4) 

8- 

-12 

2    -3 


2-5     3 


-2 


2 


Q.E.F. 


3 

2a;  — 3  is  the  h.c.msr.  sought, 
theh.c.msr.  ofa;^+3ar^4-5a;  +  3  and  x^-\-Qx^+^x-\-4. 
is  ic+l  : 

1 


a?-\-   3a;2+  5a;  +  3 


-21ic3- 


22a;3+22af 


28ar—   7x 


x+l 


3-7a; 


3a;+l 


a:3+Ga;2-f-9a;  +  4 
a;^+3a;^+5a;  +  3 
3  x^+  4  a;  +  1 


^a?+^x 


1      3      5 

3 

1    -6    -7 
22    22 

iC+l 


Q.E.F. 


1        1. 

by  83'nthetic  division,  to  find  the  h.  c.  msr.  of 
a<4.3a36+5a2  6-+5a6-^+2Z>*and2a3+5a26+4a&2+63: 

13       5       5       2 


6 


-4  -10     -8 
15     12       3 


7)   7     14 


1 


1 


-2     -4     -2 

-1     "2     -1 


+  2o6  +  6^  is  the  h.  c.  msr.  sought. 


Q.E.F. 


94 


MEASURES,  MULTIPLES,  AND  FACTOES.       [III.  pr. 


So 


to  find  the  h.  c.msr.  of 'the  numerals  679,  301 : 


679 

or            679 

602 

2 

301 

301 

2 

77 

3 

231 

77 

3 

70 

1 

70 

70 

1 

7 

10 

70 

7 

10 

Q.E.F. 


and  the  h.  c.  msr.  sought  is  7. 

Note  3.  If  either  of  the  two  numbers  be  a  product  of  known 
factors,  or  if  both  of  them  be  such  products,  the  work  in  (6)  is 
shortened  as  follows : 

Let  A  and  b  be  any  two  numbers, 
and  let    a  =  Aj •  a^  •••  a^  and  b  =  Bj  •  Sa  •••  b,», 
wherein   Ai,  Ag,  •••  are  prime  to  each  other,  and  so  are  Bi,  Bg,  •••, 
but  Ai,  A2,  •••  Bi,  Bj,  •••  are  not  necessarily  primes  or  powers 

of  primes ; 
then   •.*  every  factor  of  the  h.  c.  msr.  of  a  and  b  which  is  a 
prime  or  a  power  of  a  prime,  can  be  a  factor  of  but 
one  term  of  the  series  Ai,  Ag,  •••,  and  so  of  the 
series  Bi,  Bg,  •••, 
.•.  the  h. c.msr.  of  a  and  b  is  the  product  of  the  m-n 
h.  c.  msrs.  of  the  pairs  of  numbers  Ai,  Bi  •••,  formed 
by  combining  each  of  the  m  numbers  in  the  series 
Ai,  A2,  •••  A^  with  each  of  the  n  numbers  in  the 
series  b^,  B2,  •••  b„, 
and  these  measures  for  the  most  part  are  detected  by  simple 

inspection. 
E.g.,    if  A  =  Ai .  A2 .  A3  =  (a'  -  b') .  (aj2  _  f) .  (a'b'-  x^y') , 
and  B  =  Bi .  B2  •  B3  =  (a^  +  b^)  •  (a^  +  2/^)  •  (ab  +  xy) , 

then    •.*  Aj  and  Bj  contain  only  a  and  b,  Ao  and  B2  onlj^  x  and  2/, 
and  A3  and  B3  onlj'  factors  with  all  four  letters  a,  b,  x,  and  ?/, 

.*.  Ai,  As,  and  A3  are  prime  to  each  other,  and  so  are 
Bi,  B2,  and  B3.  [ax.  5 

And    •-•  Ai  is  also,  for  the  same  reason,  prime  to  B2  and  B3,  A2  to 
Bj  and  B3,  and  A3  to  Bi  and  Bg, 
.-.  h. c.msr.  A,B  =  h.  c.msr.  (ai,Bi)  x  h.  c.msr.  (Ag,  Bg) 
X  h. c.msr.  (a3,B3) 
=  {a-\-b)x{x-{-y)y.{ab-\-xy).    q.e.f. 


2.  §6.]     FINDING  THE  LOWEST  COMIHON  MULTIPLE.  95 

§  6.    PROCESS  OF  FINDING  THE  LOWEST  COMMON  MULTIPLE. 

PrOB.  2.     To  FIND  THE  LOWEST  COMMON  MULTIPLE  OP  TWO  OR 
MORE   NUMBERS. 

(a)    The  prime  factors  of  all  the  numbers  known: 
Multiply  together  all  the  different  prime  factors,  each  with  the  great- 
est exponent  it  has  in  any  one  of  the  numbers,    [th.  5  cr.  5,  cr.  13  nt. 
E.g.,    of  daWc-\  12a^b^d\  and  15a*  +  21a^bd,  the  different 
prime  factors,  to  their  highest  powers,  are 
22,  3\  a-,  6^  c\  d\  5a^-{-7bd, 
.-.  thel.c.mlt.  is3G-a^'b^'d*'{Da"-\-7bd), 

=  180a'b''d*-h2o'2a''bH'.  q.e.p. 

So        to  find  the  1.  c.  mlt.  of  a^  —  b^,  a^  —  &^,  and  a'^  —  b^: 
•  ..•  a'-b''  =  {a-b)'{a  +  b), 

a3  _  Z;3  =  (a  -  6). (a2  +  a6  4- &'), 
and  a*  -  6^  =  (a  -  b)  -  (a  +  b)  •  {a"  +  b'') ; 

.*.  the  I.e. mlt.  sought  is 

(a-6).(a  +  6).(a2  +  a6  +  6=^).(a2  +  &2).  q.e.f. 

{h)    The  prime  factors  not  known;  two  entire  numbers: 
Divide  the  product  of  the  two  numbers  by  their  h.  c.  msr. ;  the 
quotient  is  the  I.  c.  mlt.  sought.  [th.  5  cr.  9 

Or,  divide  either  number  by  their  h.c.msr.  and  multiply  the 
quotient  by  the  other  number. 

E.g.,    to  find  the  I.e.  mlt.  of  aj2  4- a; -12  and  iB2_i0a;  +  21: 
•.•  their  h.  c.  msr.  is  aj  —  3,  [pr.  1  (b)  ex. 

...    {x'  +  x-12)'(a^-lOx-\-21):  (x-S), 

=  {x'-^x-12).(x-7),  =a^-6x^-ldx-\-8i, 
is  the  1. c. mlt.  sought.  q.e.f. 

(c)  The  prime  factors  not  known  ;  three  or  more  entire  numbers : 
Find  the  I.  c.  mlt.  of  any  two  of  the  numbers  (preferabl}^  the  two 
highest) ;  then  the  I.  c.  mlt.  of  this  multiple  and  the  next  number, 
and  so  on,  till  all  the  numbers  are  used;  the  I.e. mlt.  last  found 
is  the  I.  c.  mlt.  sought.  '  [th.  5  cr.  6 

E.g.,    to  find  the  1.  c.  mlt.  of  289,  323,  361  : 

The  I.e. mlt.  of  289  and  323  is  5491,  \_{a) 

and  the  l.c.mlt.  of  5491  and  361  is  104329.     q.e.f.  [(a) 


96  MEASUEES,  MULTIPLES,  AXD  FACTORS.       [III.  pr. 

Note.    The  solutions  of  Pr.  1  (a)  and  Pi'.  2  (a)  extend  to  the 

h. c. rasr.  and  I.e. mlt.  of  an}-  numbers  that  are  resolved  into 

factors  prime  to  one  another,  whether  into  prime  factors  or  not. 

E.g.,'--  ^a'-lOa  +  S,    G5-  +  76-20,    m^  +  7^^  are  all  prime 

to  one  another,  [inspection 

.-.  of    (3a2-10a  +  3)3.(G62  +  76-20)2.(7?i3-(-w'*) 

and(3a2-10a  +  3)2.(662+76-20)3.(m3-f«^)"^ 
the  h.  c.msr.  is 

and  the  1.  c.  mlt.  is 

(3a2-10a  +  3)3.(662H-76-.20)3.(m'^  +  w3). 

§  7.     PROCESS   OF   FACTORING. 

PrOB.  3.       To  FACTOR  AN  ENTIRE  NUMBER. 
IN  GENERAL. 

Take  out  all  monomial  factors  by  inspection;  by  inspection  also, 
or  by  tried,  find  an  entire  measure  of  the  remaining  factor;  then 
of  this  measure,  and  of  its  co-factor;  and  so  on,  till  no  composite 
factor  remains.  Write  the  prime  factors  in  order,  and  mark  each 
one  of  them  with  that  exponent  which  shows  hoio  many  times  it 
has  been  used. 

ES'  PARTICULAR. 

(a)    Tlie  number  an  integer : 

Divide  the  number,  and  the  successive  quotients  in  order,  by 
the  pnmes  2,  3,  5,  •••,  using  each  divisor  as  many  times  as  it 
measures  the  successive  dividends.  The  successful  divisors,  and 
the  last  undivided  dividend,  are  the  prime  factors  sought. 

Note.  No  divisor  larger  than  the  square  root  of  the  dividend 
need  be  tried. 

For     *.*  dividend  =  divisor  X  quotient,  [I.  §  9  df. 

.'.  if  divisor  >  -^/dividend,  then  quotient  <  y'dividend ; 

[II.  ax.  18 
i.e.,  .        if  there  be  a  factor  larger  than  ^dividend,  there  is  also 

a  factor  smaller  than  Y/dividend, 
which       is  impossible,  since  all  factors  smaller  than  -^/dividend 
have  already  been  tried,  and  have  failed. 


3.  §  7.]  PROCESS   OF  FACTORING.  97 

Hence  every  composite  number  has  some  factor  not  larger  than 
its  own  square  root ;  and  if  a  number  have  no  such  factor  then 
it  is  known  to  be  prime. 

E.g.^    of  11908  710,  2  is  a  successful  divisor  once,  3  twice, 
5  once,  11  once,  23  once,  and  the  square  root  of 
the  quotient,  523,  is  smaller  than  23  ; 
.-.  the  prime  factors  are  2,  3,  3,  5,  11,  23,  523, 
and  11  908  710  =  2  •  32 . 5 .  11 .  23 .  523. 

(6)  The  number  a  polynomial  that  can  be  reduced  to  some 
type-form  whose  factors  are  known : 

Reduce  the  number  to  the  type-form^  and  write  its  factors  di- 
rectly^ in  the  form  of  the  factors  of  the  type. 

E.g.,    x'-\-2ax-{-a^-2om^n\  ^{x  +  aY -  (5mny, 

=  (a;  +  a  +  5  mn)  •  (a;  +  a  -—  5 mn) .  [II.  3,  2 

(c)   The  number  a  polynomial  with  one  letter  of  arrangement : 
Find  the  h.c.msr.  of  the  coefficients,  and  divide  by  it. 
By  tnal  find  a  polynomial  factor  of  degree  not  higher  than 
half  the  degree  of  the  polynomial. 

Try  no  factor  unless  its  first  and  last  coefficients  measure  the 
first  and  last  coefficients  of  the  number,  respectively. 

Ti^  no  factor  unless  its  value  measures  that  of  the  polynomial 
when  the  letters  have  convenient  integral  values  given  to  them. 

If  all  the  coefficients  in  the  polynomial  be  positive,  try  no  faxitor 
whose  first  and  last  coefficients  are  not  both  positive. 

For      no  integer  or  simple  literal  monomial  can  measure  a  poly- 
nomial unless  it  measures  ever}'  term  of  it.   Q;  e.  d. 
And     if  there  be  a  factor  whose  degree  is  higher  than  half  the 
degree  of  the  polynomial, 
then         its  co-factor  is  of  degree  lower  than  half  the  degree  of 
the  polynomial,  [II.  th.  5  cr.  5 

i.e.,  lower  than  the  degree  of  the  factor  tried, 

and  the  lower  factor,  not  the  Mgher,  is  best  sought,  q.e.d. 

And  •.•  the  ^  ^^^^  term  of  the  dividend  is  the  {  ^^^l  term  of  the 

first 
divisor  multiplied  by  the  -{  |    .  term  of  the  quotient, 


98  MEASURES,  MULTIPLES,  AND  FACTOES.        [III.  pr. 

.'.  every  entire  measure  of  the  dividend  has  its  •{  -i^^  term 
a  measure  of  the  -{  ,     ,  term  of  the  dividend,  q.  e.  d. 

And*.*  if  there  be  an  entire  measure  of  the  polynomial,  the 
co-factor  is  then  entire,  [ax.  8 

.*.  whenever  the  letters  have  integral  values 
then         the  value  of  the  co-factor  is  an  integer,  [II.  ax.  23 

i.e.,  the  value  of  the  factor  then  measures  the  value  of  the 

polynomial.  q.  e.  d. 

The  last  clause  of  the  rule  is  based  on  principles  stated  later. 
E.g. ,    to  factor  40  aa:i^  +  130  axy  -f-  75  ay^ : 

*.*  a  is  a  common  factor,  and  5  the  h.  c.  msr.  of  40,  130, 

and  75, 
.*.  the  expression  is  resolved  into  the  three  factors 
5,  a,  8x^-\-26xy  +  15f, 
wherein    1,  2,  4,  8  are  the  measures  of  8,  and  1,  3,  5, 15,  of  15  ; 
and    *.*  all  the  coefficients  are  positive, 

.*.  the  possible  measures  of  8a;^+  26icy-\-15y^,  on  its  face, 
are: 
jB  +  y,         2a;  +  2/»         4a;  +  2/^         Sx-{-y, 
x  +  3y,      2x  +  3y,      Ax  +  Sy,      8x-\-2y, 
x  +  oy,      2x-\-6y,      4:X-\-6y,      8x-\-5y, 
x-{-loy,    2ic  +  15?/,    4a;  +  152/,    Sx-^-lby, 
In        8  xr+ 26  xy+16y^  and  in  these  sixteen  possible  measures 
put  ic  =?  1    and   y  =  1  ; 

then         8of-\-26xy-{-lDy^  =  49,  whose  measures  are  1,  7,  and  49,. 
and  only  Ax  +  Sy,  =7,  and  2cc  +  5^/,  =7,   pass  this  test ; 

and  Ax  +  Sy   and    2x-{-5y   are  found  by  actual  multipli- 

cation or  division  to  be  the  factors  sought. 
So        to  factor  p,  =  7a^-  30aj2+  62a;  -  45  : 
The  onl3'  possible  linear  factors,  on  its  face,  are 

x±l,      x±3,      a;  ±5,      x±d,     a;  ±  15,      a;  ±  45, 
7a;  ±1,    7a;  ±3,    7a;  ±5,   7a;  ±  9,  7a;  ±15,  7a;  ±45. 
In         7a;^— 30a.'2+62a;— 45,  and  in  these  twenty-four  possible 
factors,  put  a;  =  1 ; 


3.  §7.]  PROCESS   OF  FACTOEING.  99 

then         p  =  —  6,  and  the  onl}'  possible  factors  of  it  are 

a;+l,=2;        a;  — 3,  =  —  2;     re +  5,  =6; 
7ic-l,=6;     7a;  — 5, =2;       7a;-9,=-2. 

So        put  x=2  ; 
then         p=15,  and  out  of  the  six  possible  factors  above  the 
only  ones  still  possible  are 
a;  +  l,=3;      a;  — 3,=-l;       7a;  — 9, =5; 
then   •.*  of  these  three  possible  factors  7a;  — 9  succeeds,  and 
gives  a;^—  3  a;  4-  5  for  quotient,  and  the  others  fail, 
.*.  a^—  3a;  +  5  is  prime  ; 
and  7a;  —  9,  a;^  —  3a;4-5  are  the  factors  sought. 

Note.    For  further  discussion  of  this  case  see  XI.  th.  4. 

(d)  77ie  number  a  polynomial ;  several  letters  of  arrangement : 

Arrange  the  number  as  to  the  powers  of  any  one  of  the  letters 
(preferably  that  one  whose  powers  are  most  numerous),  and 
unite  all  terms  having  any  same  power  of  this  letter  iiito  a  com- 
plex term.  Find  the  h.c.  msr.  of  the  coefficients  of  the  different 
powers  of  the  letter  of  arrangement^  and  take  it  out  as  a  factor 
of  the  polynomial ;  then  the  co-factor  has  no  prime  measures  free 
from  this  letter. 

Arrange  the  polynomial,  or  the  co-factor  just  found,  as  to  any 
other  letter,  and  proceed  as  before,  and  so  on  for  all  the  letters; 
of  the  co-factor  left,  the  prime  factors,  if  any,  will  each  contain 
all  the  letters,  and  can  only  be  found  by  trial;  but: 

Try  no  factor  of  more  than  half  the  degree  of  this  co-factor  as 
to  any  letter  or  letters; 

Try  no  factor  that  will  not  measure  this  co-factor  if  any  one  or 
more  of  its  letters  be  made  zero. 

If  the  polynomial  be  symmetric  as  to  any  of  its  letters,  try  no 
factor  that  is  not  either  symmetric  as  to  those  letters,  or  one  of  a 
set  of  possible  factors  that  together  are  symmetric. 

So,  if  the  polynomial  be  partially  symmetric  as  to  any  letters, 
(i.e.,  if  for  some  interchanges  among  those  letters  its  value  would 
be  unchanged,)  try  only  those  factors  which,  singly  or  in  groups, 
are  likewise  either  symmetric  or  partially  symmetric. 


100  MEASUEES,  MULTIPLES,  AND  FACTORS.       [III.  pr. 

E.g.,    to  factor  2a?-\-  Ga^y  -f  4a;/-  3x-z  +  xyz  +  2y-z  -8xz^ 

—  yz^—  3  s^ : 

'.•  2,  the  coefficieut  ofa^,  is  prime  to  6^—82;,  the  coefficient 

of  a^,  [inspection 

.*.  there  is  no  entire  measure  free  from  x. 

So   *.•  the  coefficient  of  2:^  is  prime  to  that  of  2;^,       [inspection 

.-.  there  is  no  entire  measure  free  from  z. 
But*.*  the  coefficients  of  y-,  y,  y^  have  a  h.  c.  msr.  2a;  +  2;, 
.*.  2 a;  4- 2;  is  a  factor  of  the  poknomial, 
and  the  co-factor  is  a;^-}-  3  a;^/  +  2  ?/^—  2  a;2;  —  2/2;  —  3  2;^, 

whereof  everj^  factor  has  all  the  letters,  but  reduces  to 

a  factor  of  2y^—  yz—3z-,  =  2/-f^-  2y—Sz,  when  a;  =  0, 
a  factor  of  a;^— 2 a;2;— 32;-,  =  a;-|-2;-a;  — 32;,    when2/  =  0, 
a  factor  of  a:r-\-Sxy-{-  2 y-,  =  x-\-y  •  x-\-'2y,    when  2;  =  0 ; 
and     *.*  the  trinomials  a; -}- y -h  2;,  a; +  2?/  — 82;,  and  no  others, 
fulfil  these  conditions,  and  are  found  by  trial  to 
succeed, 
.*.  the  factors  of  the  given  polynomial  are  2a;+2;,  x-\-y-\-z, 
x-{-2y  —  Sz.  Q.E.F. 

So,  to  factor  a^—2xy-^  y^—  2xz  —  2yz-\-z^: 

;  *.*   —2y  —  2z  and  if  —  2yz-i-z',  the  coefficients  of  x  and 
of  a;'',  are  prime  to  each  other, 
.*.  there  is  no  entire  measure  free  from  x ; 
so  there  is  no  entire  measure  free  from  y, 

and  none  free  from  z  ; 

.*.  every  factor  has  all  the  letters,  but  reduces  to 

a  factor  of  y^—  2 yz  -\-  2;^,  i.e.  to  ±  (y—z) ,  when  a;  =  0, 
a  factor  of  x^—2xz  +  ^-,  i.e.  to  ±  (a;— 2;) ,  when  y==0, 
a  factor  of  x^—2xy-^  ?/^  i.e.  to  ±  (x—y) ,  when  2;  =  0; 
and     *.*  no  trinomial  fulfils  all  three  conditions, 
.*.  the  given  poh'nomial  is  a  prime. 
Or, '.'  the  given  polj-nomial  is  symmetric  as  to  a?,  y,  z, 

.*.  the  factors,  if  an}^,  must  be  symmetric  as  to  a;,  y,  2;, 
either  as  a  set,  or  singly  ; 
but     -.*  such  a  set  would  consist  of  at  least  three  factors, 


3.  §  7.]  PROCESS   OF  FACTORING.  101 

and     •••  this  polynomial,  being  of  the  second  degree,  can  have 
but  two  factors, 
'.  the  factors  are  not  sj^mmetric  as  a  set. 
And    •.•  there  can  be  no  single  symmetric  factor  except  x-\-y-\-z, 
and     •.•  a:r-2xy-^y"'-2xz-2yz  +  z'^(x  +  y  +  zy, 
.  there  are  no  factors  sj'mmetric  singly, 
.  the  pol3'nomial  is  a  prime. 
Note  1.    The  proofs  for  the  rule  in  case  (cZ)  are  substantially 
the  same  as  those  given  in  case  (c) . 

Note  2.  The  work  is  often  aided  by  introducing  new  letters 
of  arrangement,  as  to  which  the  pol^'nomial  is  more  simple,  or  is 
homogeneous. 

E.g.,    to  factor  p,  =  Gx^^if—  20x^y*z^  +  25 xhfz  -Sx'^y-z^ 

-Sx'yz^-^GzK  '    '  "   '      '' 

Let       u=.x^y,  v  =  2- ,    and  seek  the  factors  of 

P,  =  6  w*  -  20  w^-v  -f-  25  u^v^  -  8  wv^  —  8  wv*  +  6  V:  ' 

Try  no  factors  except  of  the  form  ku  +  bv,  or  cw^+  Tmv-\-  E'y^, 

wherein  a,  b,  c,  e  are  measures  of  6,  and  the  value  of  the  pro- 

p§)sed  factor  is  a  measure  of  p  when  for  u  and  v  are  put  any 

convenient  integers. 

When     w, 'i;  =  l,l,     then  p  =  1,     and  a +  b  and  c +  d +  e, 

measures  of  p,  each  =  1. 
When     w,  v  =  1,  2,    then  p  =  223,  a  prime,    and  a  +  2b  and 
c  -}-  2d  +  4e,  measures  of  p,  each  of  them  =  223, 
which   is    manifestly  larger  than  the  other  con- 
ditions permit,  or  else  =  1. 
When     w, 'U  =  2,  1,   then  p  =  30,   and  2a+b  and  4c+2d+e, 

measures  of  p,  are  measures  of  30. 
But  *.*  no  integers  a,  b,  measures  of  6,  satisfy  all  these  con- 
ditions, 
.*.  there  is  no  measure  of  the  form  xu  +  bv. 
And  *.*  the  onl}^  integers  c,  d,  e  that  satisfy  them  are  2,  —4,  3, 
and     • .  •  2 1*^  —  4  ^tv  H-  3  v^  is  found  on  trial  to  measure  p,  and 
the  quotient  is  3  w^  —  4  w^v  +  2  -v^ ; 
.-.  V  =  {2u^ -  4.UV  +  ^v^)'{^u^  - ^y?v -\-2v^) 

=z{23^y''-4.Q?yz^  +^z)'{^3^y^-4.3?yz^-\'2z^). 


102  MEASURES,  MULTIPLES,  AND  FACTOES.  [IIL 

So        to  factor  p,  =  a  —  bo;  +  cx^  —  bx^  -\ ,  wherein  a,  b,  ••• 

are  positive : 

Let  —y  =  x,   then  p  becomes  a -\- By  -{- cy^  -\-  By^  H ,  whose 

factors  are  often  more  easil}'  found. 
So        to  factor  p,  =  36a;*  —  25a.'2  +  4 : 

Let  w  =  ic^,  v  =  1 ,  then  p  becomes  36  w^—  25  wv  +  4  v^,  whose 
factors  are  4w  —  i;,  9w  —  4v; 

i.e.,         4x^-1,  9x2-4,  =(2ic4-l)-(2a;-l),  (3a;+2)-(3a;-2). 
.-.  p  =  (2a;H-l).(2a;-l).(3a;  +  2).(3x-2). 
Note  3.     A  polynomial  may  often  be  resolved  into  surd  or 
imaginary  factors. 

E.g.,     x-y  =(a;^+2/5).(a;^-2/*). 

So       ,2x-3y={^2x-\-^Sy)'(^2x--^S7j) 

=  (V2a;+V32/)-«/2a;+</32/).(</2a;-^32/) 

So         ir  +  l   =(a;+V-l)-(a^-V-l)- 

§  8.     EXAMPLES. 
§§  5,  6.      PROBS.  1,2. 

•  ••  12.   Find  the  h.  c.  msr.  and  1.  c.  mlt.  of: 

1.  X"l,x^-l;x-2,a^-4:;  S(a^-a^x),4:(x^+ax),  5(x*-a*). 

2.  l-ar^,  {l  +  xY;  l-2x,  l-Ax",  l-8a^,  l-lGxS  1-32x5. 

3.  X24.2X-3,  x^-7x2+6x;  x«+aj^+a^+a^+a;+l,  x2_a.+l. 

4.  4  +  5x  +  x2,    8-2x-x2^   12  +  7x4-a;2,   20+a;  — x^^ 

5.  529(x2  +  x-6),    782(2x2  +  7x  +  3),    935(2x2- 3x- 2). 

6.  713? -\-^n7?y ^2nxy^—2nif,   Ama? -{-mo^y—2mxy^—^m'if. 

7.  X*— px^+(g— l)x2+px  — g,   x'*— gx-^4-(p— l)x2+gx— jp. 

8.  x3+(4a  +  6)x2+(3a2+4a6)x  +  3a26, 
x3+(2a-5)x2-(3a2+2a6)x4-3a2&. 

9.  aV^  +  e^^-a-^-l,    (a- 2 +  a-i).(e*- 2 +  6"''). 

10.  x2  +  2/-+;^2+2(x2/  +  2/^  +  2a;),    (Jx  +  i^/ +  2:)'-(ia;  + J2/)'. 

11.  x-«  +  fx-2  +  fx-i  +  l,   ix-2-i;  ^^-3/^        E!±^. 

x^- 2x?/  +  2/       cc  —  2/ 


§8-] 


EXA]VIPLES. 


103 


12. 


a^+  a\x  —  ab 

-b\ 


x^-\-  a\  x  —  ac 
—  c 


ar+2a 
-35 
+4c 


XT—  (yab 
H-  Sac 
-12  be 


X— 24:  abc 


x^—2a 
-3b 
+  4c 


xr-{-  (jab 
—  Sac 
-12bc 


X -{-24:  abc 


13. 
14. 
15. 

16. 
17. 

18. 
19. 

20. 


15.  Reduce  to  lowest  terms  b}^  means  of  the  h.  c.msrs.  of 
their  numerators  and  denominators  : 
a^-Gx^5        l+3a?-4a;^-12a^.         l-{-a^  +  25x'' 
■    8a^ 


7ar  —  12  a; -4-5 

r^Sa  _^  a;2a  ^  ^a  ^  1 


4a;2-2a;-fl'    l  +  3a;-15a;2-25i»^ 


x^  —  a^ 


a^+x^ 


1 


ao(^ 


,^' 


»^«+a^" 


a;-^  +  lla;-^  +  30         .    xy^^  +  2 -\- x-'^ y  .    ^_±j^^ 

-2    I     /v,-2, 


9a;-3  +  53a;-2-9a;-i— 18'        xy-^  +  x-'^y     '     a;^  +  2/"^ 

20.  Reduce  to  lowest  common  denominator,  by  means  of  the 
1.  c.  mlts.  of  the  given  denominators,  and  add  : 
1  3  5'  7  9 

2{a  +  xy  A{a-xy  Q{a'-^a^)'  Sia^-x")'  10{a+ax+x^) 
m?  -\-y^  ^—  y^  x^  -\- xy  -\- y'^  a?  —  xy  -\-y^  ^  —  y  ^-\-y 
a?  -^-y^  '  a?  -y"^  ^  x-\-y  x  —  y 
1  1  1 


01?  — if     a?-\-y^ 

1 

^x\x-^yy      2x\x'-\-y^)'     4x^{x-yy      2x\x^-y^) 

a±b       a-b       a^-hb^       a^-b^ 
a-b'      a  +  b'      a^ -V^      a-'H-^^ 


—  c  \x  —  cd 

■Vd\ 


XT  -\-a    X 
-b 


ab 


y?  —  a 


x  —  ad 


ar  —  b  \  x  —  bc 


sc^-\-b  \x-\-bd 

+  d  I 
x^  -\-a  \x-\-ac 

+  c  I 

op^  —  a  \x-\-  ac 

—  c  I 
x^  —  b  \  x-\-bd 


a^  +  b^       g'  -  W_ 
a^-b^'      o?  +  b^' 

x^  -\-b    x—bc 
—  c 


x^  -\-a 
-d 


x  —  ad 


a?  — a 
-\-b 

X- 

-ab 

x'  +  c 
-d 

X- 

-cd 

-d  I 

§  7.      PROB.  3. 

21.  Factor,  or  prove  to  be  prime  : 
30;  37;  72;  120;  323;  367;  1331 

22.  Make  a  table  of  the  prime  numbers  from  0  to  400 


1683;  8279;  15  625. 


104  MEASURES,   MULTIPLES,   AND   FACTORS.  [III. 

Note.  The  work  is  aided  b}^  arranging  the  odd  numbers  1,  3, 
5,  7,  9,  •••,  399  upon  paper  ruled  in  squares,  and  marking  off  as 
composite  every  3d,  beginning  with  3-,  every  5th,  beginning  with 
5",  etc.  The  mnlti[)les  of  any  prime,  p,  thus  marked  off,  have  a 
common  difference,  2  p,  and  often  lie  in  convenient  diagonal  lines. 
All  the  multiples  of  p  thus  got  from  one  another  ma}^  be  tested 
b}'  merel}'  testing  the  highest  of  them  b}^  division.  Why  are  the 
small  primes  most  frequent? 

23.  Use  the  above  table  to  factor  9991,  or  to  prove  it  prime. 

24.  Tabulate  the  prime  factors  of  the  numerals  1,  2,  3,  •••,  100. 
•  ••  46.  Factor,  or  prove  to  be  prime  : 

25.  iTl^  +  aT3.^+l+3a;   ccr+'2xy -\-if -]-5x-\-0  7j +  Q. 

26.  x^-^y^  +  !r  —  2xy±'2xz^:2yz;   7?  -\-'if  —  z-  ±2xy. 

27.  '2an--ir2a^c'  +  '2h-(r-a^-h^-'C\    =^a-h''-{a-+h''-cy. 

28.  a2  +  46--|-9(r  +  ...  -f- 4a&  4-6ac  +  ••• -f  126c  +  ...  +  ... 

29.  4a26-*  +  \2ah-'-(?d-^  +  ^(^d?-  lem^^Ti"^ 

If  40?7i*7i-Vg"^  -  25p"g-i«. 

30.  e^'-e--';    e''±2  +  e-'-'. 

31.  {a-a-')^-{b-b-'y;    a^-2DGb-^;    (a +  »)«-(a-aj)«. 

32.  aV-3a3a;  +  2a^   a^-a^x- Gax" ;    12a*+a^x^-x\ 

33.  a^  +  ^  4- 3 a;?/ (a;  +  2/)  5    m^ —  n^  ^m(m^  —  7r)-\-n{m~7i)^. 

34.  a^-ab-  2{ab  -  b')  +  3(a2  -  b')  -  4(a  -  by, 

35.  a^-b^-Sab{a-b);    b  (x^ -y')  +  3{x  +  tjy, 

36.  S{a^-f)-5{x-yy;    (x  +  yy +  2{x' +  xy)-S(x^ -f). 

37.  2{a^-\-a'b-\-ab')-(a^-b^);   a*-b*+{a^-by. 

38.  2a^y-\-oxry^-{-2x7/;    ey*-Sxf-9aPf;    Q> x"^ -\- x" y -\2 y\ 

39.  a'^x'-^-a^x-a?',  Gb''x--7bxr-Sx*;    6ar^+ H  a^+9a;-35. 

40.  ear''— liar -j- 9 a; +  35;    6ar^— 11  a^+ 9 a; +  34. 

41.  x^-{-{a  —  b  -{-  G  —  d)x^-\-  (—  ab  -\-  ac  —  ad  —  bc  +  bd  —  cd)x^ 

+  (  —  abc  +  abd  —  acd  +  bcd)x  +  abed. 

42.  3a;^-17ar^  +  38ar-23a;  +  9;    5a;^-18a;3^  i7^,2_5^.p^  5  . 

15x*-}-8a^y  —  S2  xy^  —15y*.  [two  trinomial  factors 

43.  abx'+a^x  +  b'^x-^ab;   a^a^-\-bY-\-(^^;   a^x^-b'f +  c^^. 


§  8.]  EXAMPLES.  105 

44.  39(a^+a36«)a;^2/'+78(a«+2a363+6«)a^/4-156(a363+5<5)a;2y. 

45.  3 a-+  6 ahz  —  4 ac2;  —  8  hcz^ ;    3  a?—  6  ahz  +  4 aca;  —  8 6cs^. 

46.  aV4-2a2  6a;2_^2a62a;4.63.   a^x^+a'^ly'x''+h\ 

47.  Factor  45a^+83ar> -100x2/2-492/3^ 

Note.    Only  measures  of  the  form  a.x  +  b?/  need  be  tried,  and 
here  a,  b,  being  measures  of  45,  49,  are  odd  integers  ; 
but    •."  A  +  B,  the  value  of  the  proposed  measure  when  ic,  y 
each  =  1,  is  an  even  integer  and  cannot  measure 
the  corresponding  value  of  the  polynomial,  the  odd 
integer  45  +  83  — 100  —  49  ; 
.*.    the  pol3'nomial  is  a  prime. 

48.  Show  in  like  manner  which  are  primes  of: 

l7?-\0x^-\-^x  +  b,    7ar^-25a^+lla;  +  3, 
5ar^+17a;  +  3,   ar±ah  +  h\   o?±orh +  ah''±h^', 
and  generalize  for  an}'  trinomial  or  quadrinomial  whose 
first  and  last  coefficients  and  one  other  are  odd. 

49.  Resolve  into  three  symmetric  factors  : 

—  2  ar'— 2  ?/^— 2  z^  -\-b  x^  y  -\-b  y-  z+6  z^  x—xif —yz^ —zo?  -\-2  xyz. 

50.  If  ¥{x)  be  an}'  entire  function  of  a;,  prove  that  F(aj)— F(a) 

is  measured  b}'  x  —  a^  and  hence  that 
If  F(a)  =  0,  then  v{x)  is  measured  by  x—a.    Hence  factor : 
(a^+2a;  +  3).(a3+a)-(a3+2a4-3)(«3+a;). 

51.  Prove  that   yfl  y" -\- y'^  z""  ■\- z'^  x^  —  x"" y"^ —y""  z'^ —z^ xP^  is  measured 

by  {x  —  y)'{y  —  z)'{z  —  x)  if  q  and  r  be  any  positive 

integers.     Hence  factor : 

Qi?y  +  y^z  4-  z^x  —  xy^  —  yz^  —  zx^  ;• 

^y  -\-y^^  -{-z^x  —  xi/  —y^  —zx^'j 

3?y'^-\-  y^z^  +  ^y^—  o^y^—  y^^—  z^a?, 

52.  Prove  that  the  {  i    ^'  ™,!^'  of  two  or  more  numbers  is  the  re- 

•  1.  c.  mit. 

ciprocal  of  the  -(  Z  ^*  ™   '  of  their  reciprocals. 
^  '  n.  c.  msr. 

53 .  Prove  that  the  A  """^^i'^^^^     of  a  simple  fraction  in  its  lowest 

'  denominator  ^ 

terms  is  the  l.c.mlt.  of  ^  J^^  redprTcal  fraction  ^^^  ^' 


106  PERMUTATIONS  AND  COMBINATIONS.  [IV.  th. 

IV.    PERMUTATIONS  AND   COMBINATIONS. 

§  1.     DEFINITIONS. 

The  different  orders  in  which  several  things  or  elements  can 
be  put,  are  theu'  permutations  or  arrangements;  the  different 
groups  that  can  be  made  of  them,  without  regard  to  order,  are 
their  combinations.  Two  permutations  are  different  when  either 
the  things  themselves  are  different  or  their  order  of  arrangement 
is  different;  but  two  combinations  are  different  only  when  at 
least  one  of  the  things  contained  in  one  of  them  is  not  found  in 
the  other. 

E.g.,   ah,  ha,    ac,  ca,     he,  ch     are  the  six  permutations  of 
a,  h,  c,  taken  two  at  a  time  ; 
but  ah  and  ha  are  the  same  combination,  ac  and  ca  are  the 

same,  and  be  and  ch  are  the  same, 
and,         in  all,  there  are  but  three  distinct  combinations. 

So        dbc,  bac,   acb,  cab,   bca,  cba  are  the  six  permutations  of 
(4,  6,  c,  taken  all  together ; 
but  there  is  onl^-  one  combination,  in  whatever  order  the 

three  things  are  taken. 
So        of  four  things,  a,  6,  c,  d,  there  are  four  combinations, 
taken  three  at  a  time  :    abc^  abd,  acd,  bed, 
and  of  each  of  them  can  be  made  six  permutations,  as  above 

—  twenty-four  in  all. 

§  2.     PEEMUTATIONS. 

PrOB.  1.  To  FORM  THE  SEVERAL  PERMUTATIONS  OF  U  THINGS, 
ALL   DIFFERENT,   TAKEN    1,   2,  3,  •••  AT  A  TIME. 

To  each  of  the  n  things  in  turn,  annex  each  of  the  (n  — 1)  things 
remaining ;  the  results  are  the  couplets. 

To  each  of  the  coiiplets  in  turn,  annex  each  of  the  (n— 2)  things 
remaining ;  the  results  are  the  triplets. 

To  each  of  the  triplets  in  turn,  annex  each  of  the  (n  — 3)  things 
remaining ;  the  results  are  the  fours;  and  so  on. 


1.  §  2.]  PERMUTATIOXS.  107 

E.g.^    of  the  four  things  a,  &,  c,  d  the  permutations  are : 
sino;le  thinsfs : 


a, 
couplets : 

a6,  ac,  ad, 
triplets : 

ahc^    abd, 

6a,  be,  bd, 
bac,    bad, 

ea,  eb,  cd, 
cab,    ead, 

d; 

da,  dh,  do; 
dab,   dac. 

acb,    acd, 

bca,    bed, 

cba,    cbd, 

dba,    dbe. 

adb,   ode, 
fours : 

abcd^  abdc, 

bda,   bdc, 
bacd,  bade, 

cda,    edb, 
cabd,  eadb. 

dca,  deb; 
dabc,  dacb, 

acbd,  acdb, 

bead,  bcda, 

cbad,  cbda. 

dbae,  dbea, 

adbc,  adcb^ 

bdac,  bdca. 

edab,  cdba. 

dcab,  dcba. 

Theor.  1.     The  proeess  of  Pr.  1  gives  all  the  possible  jyermu- 
tations,  and  no  two  of  them  so  formed  are  alike. 

The  number  of  permutations  ofn  things,  all  different, 


taken 


1 
2 

^  at  a  time,  is 

r 


n. 


n.(n-l). 
n.(n-l).(n-2). 


n.(n-l)-(n-2)...(n-r-f-l). 
Let  a,  b,  c,  cZ,  •••  h,  Tc,  I  =  any  n  things  all  different,  and  let 
T^n,  T.^n,  PgW,  •••  p^n  =  the  number  of  permutations  of  these  n 
things  taken  1  at  a  time,  2  at  a  time,  3  at  a  time,  •••  ?*  at  a  time. 

(a)    One  at  a  time. 
For    ;.•  of  one  thing  there  is  one  and  but  one  permutation, 

.-.  of  the  n  things,  taken  one  at  a  time,  there  are  n  per- 
mutations, viz.,  one  for  each  thing,  and  no  more. 
i.e.,         Fin  =  n.  q.e.d. 

(6)   Tivo  at  a  time. 

For      to  each  of  the  n  things  in  turn,  annex  each  one  of  the 
n  —  1  things  remaining ; 
then         ab,  ac,  ad,  •"  al  form  (n—1)  couplets  with  a  first, 
ba,  be,  bd,"-  bl  form  (n  —  1)  couplets  with  b  first, 
ca,  cb,  cd,  '"  cl  form  (n  —  l)  couplets  with  c  first. 


and  la,  lb,  Ic,   ---Ik  form  (n—1)  couplets  with  Z  first; 

whereof  no  two  are  alike,  since  each  one  has  either  a  first  letter, 
a  second  letter,  or  both,  different  from  everj^  other. 


108  PEKMUTATIONS   AISTD   COMBINATIONS.  [IV.  th. 

and  there  are  no  possible  couplets  omitted,  since  eveiy  letter 

in  turn  is  joined,  both  as  first  letter  and  as  second 
letter,  with  ever}'  other  letter ; 
.*.  of  the  71  things,  taken  two  at  a  time,  there  are  w  •  (n  —  1) 
permutations,  and  no  more  ; 
i.e.,  Pan  =  71- (71  —  1).  Q.E.D. 

(c)   Tliree  at  a  time. 

For      to  each  of  the  7i-(7i— 1)  couplets  in  turn,  annex  each 
one  of  the  (n  —  2)  things  remaining  ; 
then   •••  ahc^  abd,  abe,  •••  abl  form  n  —  '2  triplets  with  ab  first, 
acb,  acd,  ace,  "•act  form  7i  —  2  triplets  with  ac  first, 

and  Ika,  Ikb,  Ike,  -"Ikh  form  w  —  2  triplets  with  Ik  first, 

whereof  no  two  are  alike,  since  each  one  of  them  has  either  the 
leading  couplet,  or  the  letter  that  follows  it,  or 
both,  different  from  every  other, 
and  there  are  no  possible  triplets  omitted,  since  every  pos- 

sible couplet,  in  turn,  is  followed  by  every  letter 
not  alread}'  in  it ; 
.*.  of  the  n  things,  taken  three   at  a  time,  there   are 
7i'(7i  —  l)'(7i  —  2)  permutations,  and  no  more  ; 

i.e.,  P37l  =  W«(7l  — l)-(7l  — 2).  Q.E.D. 

So  P47l  =  7l.(7l  —  l)-(7l  — 2)-(7l  — 3), 

P5n  =  71.(71  —  l).(7i  —  2).(7i  — 3)-(n  — 4), 

*"? 

and  p^/i  =  7i-(7i  —  l)'(7i  — 2)---(7i  — 7-  + 1)^  for  any  value 

of  r  not  greater  than  n.  q.  e.  d. 

Note  1.  This  proof  is  by  induction,  but  it  is  of  so  simple  a 
character  that  it  need  not  be  put  in  the  formal  order  given  in  II. 
§  1(c),  II. th,  3(c),  •••.  The  reader  may,  however,  as  an  exercise, 
make  the  statement  formal. 

Note  2.     The  expressions 
n,  7i.(w— 1),  7i-(7i— l).(7i— 2), ...  w.(7i— l).(n— 2)...(7i— r+1) 
may  be  severally  written  in  the  equivalent  forms : 

n !  n !  n !  7^ !  nl 

(w-l)!'    (n-2)!'   (71-3)!'    (7i-4)!'  *"   (n-r)l' 


1.  §2.1  PERMUTATIONS.  109 

Cor.  1.    Of  n  things^  all  different^  taken  all  together^  there  are 
n !  permutations. 
For     •. •  w  —  r  -f- 1  =  1     when  r  =  n, 

.-.  p„n  =  7i.(n  — l)-(?i-2)...3.2.1 
=  1.2-3...n 

=  n\  Q.E.D. 

Note  1.  The  expression  n  !,  hitherto  defined  as  the  continued 
product  of  the  natural  numbers  1  •  2  •  3  •  •  •  ?i  [I.  §  8] ,  may  have  a 
useful  extension. 

For     •.•  n!  =  1.2.3...n,  and  (n  -  1)  !  =  1 .2-3  •••  (71- 1), 
.*.  n!  =(71  —  1) !  -n, 
.*.    {n  —  \)\  =  n\  :  n. 
So  (n-2)!=(7i-l)!  :  («-l), 

(n— 3)!  =  (n-2)!  :  (n-2),    and  so  on. 
Conceive  this  relation  to  hold  true  for  all  integers,  whether 
positive,  zero,  or  negative  ; 
then         1!  =  2!:2  =  1,     0!  =  1!:1  =  1. 

With  this  explanation  the  form  n !  :  (?i  —  r) !  becomes  intel- 
ligible when  r  =  n,  as  in  Cor.  1,  for  then 
n\  :  (71  —  ?•)!=  n  !  :  0  !  =  ?i !  :  1  =  n  ! , 
and  the  result,  the  value  of  p^t?,  is  the  same  through  which- 

ever form  it  is  reached. 
Note  2.  Another  and  independent  proof  of  Cor.  1  is  as  follows  : 
Let  a,  6,  c,  •••  be  n  things,  all  different ; 
then   • .  •  of  the  one  thing  a,  there  is  one  permutation,  and  but  one, 
. *.  Pi  1  =  1 ,     which,  for  conformity  with  what  follows,  may 
be  written  1  !  q.e.d. 

Place  6  in  each  of  the  only  two  possible  positions  with  respect 
to  a,  i.e.  after  a  and  before  a,  giving  ah  and  ha  ; 
then   •.*  of  two  things  a,  6,  there  are  two  permutations,  and  but 
two, 
.*.  P2  2  =  2,    which  may  be  written  2  !  q.e.  d. 

Place  c  in  each  of  the  only  three  possible  positions  with  respect 
to  a  and  h  in  these  two  couplets  a6,  6a,  giving 
ahc.  acb,  cab,  hoc,  bca,  cba ; 


110  PERMUTATIONS  AND   COMBINATIONS.  [IV.  tli. 

then   •••  of  three  thmgs  a,  6,  c,  there  are  2-3,  =  6,  permuta- 
tions, and  but  6, 
.-.  P33  =  2!.3  =  3!  Q.E.D. 

Place  d  in  each  of  the  onlj-  four  possible  positions  with  respect 
to  a,  b,  c  in  these  3  !  triplets  ; 
then    •.•  of  four  things  a,  6,  c,  d,  there  are  3  !  •  4  permutations, 
.-.  P44  =  3!.4  =  4!  Q.E.D. 

So        P55  =  4!.5  =  5! 

Pg 6  =  5  !  •  6  =  6  !,     and  so  on. 
.-.  T^n=  {n—  1) ! .  71  =  71 !,     n  any  positive  integer,  q.  e.  d. 
This  note  embodies  a  rule  for  forming  the  permutations  of  n 
things  1,1  km  all  together.     The  reader  may  state  it,  and  illus- 
trate it  by  the  permutations  of  a,  6,  c  and  of  a,  6,  c,  d. 

Cor.  2.     Pr(n +l)  =  Prn-f  r.p,_in. 
For     •.•  pXw+1)  =  (7i+1). 71.(71-1).. .(?!-?'-}- 2), 
and     •.*  v^n-\-r'Vr.in  =  n'{n  —  \)"'{n  —  r  +  2)'{n  —  r-\-l) 

+  n-  {n  —1)  ...  (?i  — 7'  +  2).r 
=  71.(71— l)...(7i  — ?-  +  2).(?i  +  l), 

.*.    P^(?l-f  1)  =P^7l+r.P^_l7l.  Q.E.D. 

XoTK.  Another  and  independent  proof  of  Cor.  2  is  as  follows : 
Let  «,  6,  c, ...  A'  be  an}^  n  things,  all  different,  and  I  another ; 

then  •.*  p^7i  =  the  number  of  permutations  of  the  n  things, 
a-'-k^  taken  r  at  a  time, 

and  *.*  p^_i?i  =  the  number  of  permutations  of  the  n  things, 
a"'k,  taken  7'  —  1  at  a  time, 

and  •.*  no  permutations  of  the  7i  +1  things,  a '"I,  taken  r  at 
a  time,  can  be  formed  except  those  of  the  n  things, 
a  ...  Z:,  taken  r  at  a  time,  and  those  of  the  n  things, 
a  '"  k,  taken  r— 1  at  a  time,  with  the  new  thing  I 
placed  in  each  of  the  r  possible  positions  therein, 

.-.    P^(7l+l)  =  P^7i  +  r.P^_l7l. 

This  note  embodies  a  new  rule  for  forming  the  permutations 
of  n  things  taken  r  at  a  time.  The  reader  ma}^  state  it.  It  also 
sers^es  to  interpret  the  formula,  i.e.,  to  show  what  property  of 
the  arrangements  the  formula  expresses. 


2.  §2.]  PERMUTATIONS.  Ill 

Many  algebraic  results  derive  their  chief  interest  from  thus 
admitting  proofs  of  two  kinds,  b}'  interpretation,  and  by  more 
formal  methods  ;  and  the  two  lines  of  proof  often  curiously  cor- 
respond. The  reader  should  therefore  accustom  himself  to  look- 
ing for  such  interpretations.  He  will  find  many  of  them  connected 
with  the  subject  of  permutations  and  combinations  :  e.g.^  [th.  3 
cr.  1  nt.,  cr.  2  nt.]. 

Theor.  2.    If  n  things,  whereof  p  things  are  alike,  q  things 

n' 
alike,  r  things  alike,  •••  6e  taken  all  together,  there  are  —- -^ — ; — 

different  permutations  of  them.  i'  •   h  • 

E.g.,   if  there  be  two  5's,  three  6's,  and  four  7's,  then  of  these 

9  ' 

nine  dibits '- ,    =  1260,  different  nine-fig- 

2  !.3  !-4! 

ured  numbers  can  be  formed. 
For      take  the  n  things  in  the  several  positions  they  hold  in 
any  one  of  their  permutations,  and  let  p  things 
alone  change  places,  while  the  7i  —  p  things  re- 
maining stand  fast ; 
then         if  the  p  things  be  all  different,  p  !  permutations  are  got ; 
but  if  the  p  things  be  all  alike,  only  one  permutation  is  got. 

So        for  every  set  of  positions  in  which  the  p  things  stand. 
.-.  there  are  p\  times  more  permutations  of  the  n  things 
when  any  p  of  them  are  all  different  than  when 
those  p  things  are  alike  ; 

i.e.,  P„  W  all  different  =  P  •  *  P«  ^  p  alike- 

So  r„  n  all  different  =  P  !  '  ^  !  *  ^  !  •  •  •  •  P„  W  ^  alike,  q  alike,  r  aUke, ...  » 

But-.-    P„nyidifferent  =  '«'J 

n!  ^  ^  ^ 

•••    Pn  ^p  alike,  7  alike,  r  alike,...—   ^  J  .  ^y  I  .  ^  J  ...  *  Q- E.  D. 

In  particular  : 

^n'^n  alike      ^^  ^  ? 
-      Pn^n-lalike=W5 

n\ 

I*n^  2  alike,  n-2  alike 


2!.(n-2)! 

Pn'*  r  alike,  n-r  alike  —  ^  ]  ,  ^^  _  ^y' 


112  PERMUTATIONS  AND  CO:\IBINATIONS.  [IV.  th. 

§  3.     COMBINATIONS. 

PROB.  2.  To  FIND  THE  SEVERAL  COMBINATIONS  OF  71  THINGS, 
TAKEN  1,  2,  3,   •••  AT  A  TIME. 

To  each  of  the  n  things,  in  turn,  annex  each  of  the  things  that 
follow  it;  the  results  are  the  couplets. 

To  each  of  the  couplets,  in  turn,  annex  each  of  the  things  that 
follow  all  its  elements;  the  results  are  the  triplets;  and  so  on. 

E.g.,    of  the  four  things  a,  b,  c,  d  the  combinations  are  : 


single  things : 

a,                        6, 
couplets : 

ab,  ac,  ad,         be,  bd, 
triplets : 

abc,  abd,  a^d,     bed. 

c,                d; 

Cd', 

So        of  210,  the  prime  factors 

are :  2,  3,  5,  7  ; 

the2ddegreefactors:2.3,  2-5,  2-7,  3-5,3.7,  5-7; 
the  3d  degree  factors  :  2.3-5,  2.3-7,  2.5-7,  3-5-7. 
By  this  process  are  formed  all  the  possible  sets  in  which  the 
several  things  are  arranged  in  their  normal  order ;  viz. :  every 
such  couplet  possible,  and  from  these  couplets,  every  such  triplet 
possible,  and  so  on  ;  and  the  sets  so  formed  embrace  every  pos- 
sible combination. 

For  if  an3^  sets  were  formed  with  the  order  of  the  letters 
changed,  such  sets,  though  different  permutations, 
would  be  but  the  same  combinations  repeated.  [§  1 

Theor.  3.    The  number  of  combinations  ofn  things,  all  differ- 
ent, taken  r  at  a  time,  is 


r!-(n-r)! 

For      take  an}'  r  of  the  n  things,  and  let  them  change  places; 
then    •.•  of  these  r  things,  there  are  r !  permutations, 
but  only  one  combiuation, 

and  so  for  every  set  of  r  things  ; 

.*.  there  are  r  !  times  more  permutations  ofn  things,  taken 
r  at  a  time,  than  there  are  combinations. 
I.e.,  •Pfn  =  r\-c,n, 


3.  §3.]  COMBINATIONS.  113 

wherein    c^?i  =  the  number  of  combinations  of  «  things  taken  r 
at  a  time. 

But     •.•  P^n  =  -- -, 

(ti  —  r) ! 

n\ 

•••    CrW  =  -r-7 -.'  Q.E.D. 

Note  1.     A  useful  way  of  writing  the  formulae  is : 
n  n(n  —  Vi 

=  Y7'        ^2^  =       2!      ' 

3  !  r ! 

Ai.  71  ?i7i  —  1  n  —  1 

or  this :    Ci?i  =  -i       Con  =  -' ,    =Cin« , 

112  2 

71     71  —  In  — 2      ^       71  —  2 

Co  71  =  -  •  .  =  Co  71 


'*•■-!         2  3     ""''-       3    ' 

c,™  =  !?  .  ?L^  .  !ill2...«-r  +  l^  n-r  +  l^ 

1         2  3  r  r 

wherein  the  successive  terms  of  the  series  are  got  by  multiplying 

the  preceding  terms  by  fractions  of  the  form     ~    , 
^  °  ^  fc+1 

whose  numerators  decrease,  and  denominators  in- 
crease, by  one,  at  every  step. 

COR.    1.        C^n  =  C„._rn  =  V^HraUke,  n-taliU' 

n  ^ 

For     *.*  each  of  them  = ^ , 

r\  (n  —  r) ! 

.'.  they  are  equal  to  each  other.  [II.  ax.  1 

In  particular : 

CqTI  =  C„7l      =  1 ,  =  P„7i  „aiike- 
CiTl  =  C„_i7l  =  71,  =  P„7l„.,^ike- 

n ! 

C^n  =  C„_2  71  =  — — — — ,   =  P„7l  2alike,  n-2alike« 

2  !  (71  —  2)  ! 

_         _       '^*J         _ 

CgTl  —  C„_37l  —  — — — — ,    —  Pn'^  3  alike,  n-3  alike* 

Note.    Anotlier  and  independent  proof  of  Cor.  1  comes  from 
its  interpretation,  and  is  as  follows  : 

•.•  for  every  set  of  r  things  taken  out  of  n  things  there  is 
left  one  set  of  ti  —  r  things,  and  but  one. 


114  PERMUTATIONS  AND  COMBINATIONS.         [IV.  ths. 

.*.  the  number  of  combinations  of  an}-  n  things,  when  taken 
r  at  a  time,  and  when  taken  n  —  ?•  at  a  time,  is  the 
same.  q.e.  d. 

So        if  an}'  same  n  tilings,  whereof  r  are  alike  and  n  —  r 
alike,  be  permuted  in  any  same  n  places, 

then   *.•  when  any  two  of  the  permutations  are  -J  oKi^g       '  ^^^ 

r  things  occupy  •{  , ,    ^  ^^  ^    combination  of  places, 
®  ^-^  •  the  same  ^ 

.*.  the  number  of  permutations,  ?« ^r  alike,  n-r  alike 

equals  the  number  of  combinations,  c^n.      q.e. d. 
Cor.  2.     c,(n  + l)  =  Crn  +  Cr_in. 

For     •.•  c^wH- l)  =  i— I— £ 5^ i — 1 ! — i, 

r ! 
and     •.•  c^n  -|-c^_iW 

_7i.(?i  — l)...(?i  — r-f  1)      7i.(n  — l)...(n  — r+2) 


r!  '  {r-l)l 

__n-'(n— r+2).(n--r+l)      71  •••  (?i  —  r+  2).r 
""  Ti  "^  r] 

n.(?i-l)--.(7i-r  +  2).(n  +  l) 
= H  ' 

.-.    C^(wH-l)  =  C^n +  €(,._  1)71.  Q.E.D. 

Note.   Another  and  independent  proof  of  Cor.  2  is  as  follows  : 
Let  a,  6,  c,  •••  A;  be  an}-  n  things  all  different,  and  I  another ; 
then   • .  •  c^  71  =  the  number  of  combinations  of  the  7i  things  a-^-k, 

taken  r  at  a  time, 
and     •.•  c,._i7i=the  number  of  combinations  of  the  7i  things 

a  •••  A:,  taken  r  —  1  at  a  time, 
and     •.•  no  combinations  of  the  7i  -f-1  things  a«"  Z,  taken  r  at  a 
time,  can  be  formed  except  those  of  the  7i  things 
a-'-Jc,  taken  r  at  a  time,  and  those  of  the  n  things 
a-"k,  taken  r  —  1  at  a  time  and  followed  by  the 
new  thing  Z, 
.'.  c^(n  +  l)  =  c^7i  +  c^_in. 
This  note  embodies  a  new  rule  for  forming  the  combinations 
of  71  things  taken  ?•  at  a  time.    The  reader  may  state  it.    It  also 
serves  to  interpret  the  formula,  and  show  what  property  of  the 
combinations  the  formula  expresses. 


3, 4.  §  3.]  COMBmATIOKS.  115 


Theor.  4.    If  there  he  n  tilings,  all  different,  P?  q?  i*?  •••  he  any 

nuinhers  such  that  p+q+rH —  =  n,  then  there  are ^ 

p !  •  q  !  •  r  !  •  •  . 
ways  in  ivhich  these  n  things  can  be  made  up  into  sets,  whereof 
the  first  set  contains  p  things,  the  second  set  q  things,  the  third  set 
r  things,  and  so  on. 

E.g.,    ten  soldiers  ma}^  be  formed  into  three  guards,  of  2,  3, 

and  5  men  respectively,  in  — '——,  —  2520,  different  ways. 

For  let  the  first  p  things  constitute  the  first  set,  the  next  q 
things  the  second  set,  and  so  on,  and  let  the  n  things  change 
places  in  every  wa}'  possible,  forming,  in  all,  n  !  permutations ; 
then   •."'  within  each  set  of  p  things  there  are  p  !  permutations, 

within  each  set  of  q  things  q !  permutations,  and 

so  on, 
and     •.*  each  of  the  p  !  permutations  combines  with  each  of  the 

q !  permutations,  so  that  each  of  the  double  sets 

gives  p\-q\  permutations,  and  so  on ; 
.  • .  for  every  wa}'  in  which  the  sets  are  made  up  there  are 

p\-q\'r\"'  permutations, 

i.e.,         Pn^i=pl-g!-^'l---c^,5,r,...'^; 

•••    ^p,<i,r,..n=— -—'  Q.E.D. 

p\.q\  >r\'-' 
Note.    Expressed  in  the  notation  of  this  theorem, 

Cor.  1.  If  the  number  of  sets  be  given,  the  greatest  possible 
value  o/ Cp,  q  ,.,...  n  is  when  no  two  of  the  numbers  p,  q,  r,  •••  differ 
by  more  than  a  unit,  one  from  the  other. 

Yov,\f  p>q  +  l, 
then    '.'  p\'q\=p'{p—\)\'q\ 
and  (i)-l)!.(g-M)!  =  (g  +  l).(^-l)!.^!, 

.'.p\.q\>{p-l)\.{q  +  l)\ 

n\  n\ 


p\'q\-r\-"    ^  (|)_l)!.(^-f.l)t.r 


116  PERMUTATIONS   AND   COMBINATIONS.        [IV.  ths. 

and  c^.ff.r,...^  is  not  the  greatest  possible  if  p  exceeds  q  hy 

more  than  a  unit. 
So        of  an}'  other  pair  of  them. 

.  • .  c,, ,,  r, ...  '^  is  greatest  when  etc.  Q.  e.  d. 

In  particular :  If  n  be  an  ^     -, -,    number, 

then         c^  n  is  greatest  when  r  =  ^  I  ( 71  ±  1 V 

Cor.  2.     Tliere  are  ——— — /\\„   ,    ,,.      ways  of  making  up 
a!.b!.--(p!)*-(q!)''--- 

n  things,  all  different,  into  a  collection  of  a  sets  of  p  things  each, 

b  sets  of  q  things  each,  and  so  on;  wherein  ap  +  bq  H =  n. 

E.g.,  a  boat-club  of  10  men  can  be  divided  into  three  pair- 
oars  and  a  four  in  3  , .  ^  ,  .(2  !)«.  (4  !)^  ^  ^^^^'  ^'^' 
ferent  ways. 

For     •••  there  are  — ; : '- ; — -— wa3'S  of 

p\'p\"'a  tnnes-g!-g!-«-6  times ••• 

making  up  n  things  into  sets,  whereof  the  first  a 
sets  contain  p  things  each,  the  next  6  sets  contain 
q  things  each,  •••,  [th. 

and  •.•  of  these  wa3-s,  by  reason  of  the  permutation  of  the  a 
sets  among  themselves,  the  b  sets  among  them- 
selves, •••,  there  are  al-bl-"  for  ever}-  way  in 
which  the  collection  of  a  +  6  H sets  is  made  up, 

.'.  — =  a  !  •  6  !  •••  times  the  number  of  ways 

(i>  !)■■•(?!)'••• 

in  which  the  collection  can  be  made  up  ; 

n  ' 

.-.  that  number  is  — --r- ; — '  „  ,    .., —       q.e.d. 

al-bl-"  (piy-iqiy  — 

Theor.  5.  If  there  be  n  sets  of  things,  containing  p,  q,  r,  ••• 
things  respectively,  and  if  combinations  of  n  things  be  made  up 
by  taking  one  thing  from  each  set,  then  the  number  of  such  com- 
binations is  p  •  q  •  r  •  •  • . 

For,  let  the  n  sets  be  aj,  ag,  ag,  •  •  •  a^,   b^,  b^,  &3,  •  •  •  b^,   Ci,  Cg,  C3,  •  •  •  c^, 
•  ••,  and  write  the  first  combination  ai^iCi---  ; 
then         while  the  biC-^---  stand  fast,  substitute  a2,  a^,  -•-  a^  in 
turn  for  cti,  thus  forming  p  combinations. 


5,6.  §3.]  COMBINATIOXS.  117 

So        in  each  of  these,  in  turn,  substitute  h^^  63,  •••  h^  for  6,, 
thus  forming  q  combinations  from  one  of  them,  and 
p  -  q  combinations  from  all  of  them. 
So        in  each  of  these,  in  turn,  substitute  Cg,  C3,  •••  c^,  for  Ci, 
thus  forming  r  combinations  from  one  of  them,  and 
p-q-r  combinations  from  all  of  them. 
So        •••,  thus  forming  p-q-r---  combinations,     q. e.d. 
Cor.  1.     If  there  be  a  set  of  p  things,  a  set  of  q  things,  a  set 
of  T  things,  •••,  there  can  be  made  up  Cip-Cjq-Ckr  •••  combina- 
tions by  taking  i  things  from  the  first  set,  j  things  from  the  second 
set,  k  things  from  the  third  set,  and  so  on. 

Cor.  2.  With  the  data  of  Cor.  1  the  number  of  permutations 
is  (i-f-j+k  +  ...)!-CiP-Cjq-Ckr.-.. 

Theor.  6.  If  there  be  n  numbers,  all  different,  and  if  all  pos- 
sible homogeneous  products  of  the  rth  degree  (combinations  with 
repetition)  be  made  of  them,  including  their  rth  poivers  and  the 
products  of  their  1st,  2d,  3d,  ---  (r  —  l)th  powers  combined  in  all 
possible  ways,  so  that  there  shall  be  r  factors  in  each  product,  and 
no  more,  then  the  number  of  such  products  is 

c  n    -c  rn+r-n       Mn+D-jn+v-l) 

For,  let  a,  b,  c,  ---  be  n  numbers,  all  different,  and  in  each  of 
these  c^,  ^^h  repetitions  ^  products  let  the  letters  be  put  in  alphabeti- 
cal order,  e.g.,  aaa---,  bdde---;  and  then,  while  the  first  letter 
in  each  product  stands  fast,  let  the  second  letter  be  replaced  by 
the  letter  next  after  it  in  the  alphabet ;  the  third  letter,  by  the 
letter  next  but  one  after  it  in  the  alphabet ;  •  •  •  the  rth  letter, 
by  the  letter  that  is  r  —  1  steps  be3'ond  it  in  the  alphabet,  e.g., 
aaaa---  by  abed---,  bdde---  by  befh--- ; 
then   •.•  each  of  the  c^^^ith  repetitions^  products  is  thus  changed  into 

a  combination  wherein  no  two  elements  are  alike, 

and  no  element  is  beyond  the  (n  +  r  —  l)th  letter 

of  the  alphabet, 
.-.  each  product  is  changed  into  some  one  of  the  cXn+r—1) 

combinations  of  r  letters,  without  repetitions,  of 

(n  +  r— 1)  letters ; 


118  ^     PERMUTATIONS  AND  COMBINATIONS.  [IV.  th. 

and  •••  all  the  combinations  so  formed  are  unlike,  either  in  theii 
first  letters  or  in  their  second  letters  or  •••,  in  the 
same  way  as  are  the  products  from  which  they 
were  got, 
.'.  to  each  of  the  products  there  corresponds  a  different  one 
of  the  combinations  of  n-\-r—l  things  taken  r  at 
a  time  without  repetitions  ; 

•*•    C^,  with  repetitions  W     >     C^Ol-f-r  — 1). 

Again,  let  the  elements  of  each  of  the  c^(n  +  r  —  1)  combina- 
tions be  put  in  alphabetical  order,  and  then,  while  the  first  ele- 
ment in  each  combination  stands  fast,  let  the  second  element  be 
replaced  by  the  letter  next  before  it  in  the  alphabet ;  the  third 
element,  by  the  letter  two  places  before  it  in  the  alphabet,  and 
so  on; 

then  *.•  each  of  the  Cr(n-|-7*  — 1)  combinations  thus  gives  a 
product  wherein  no  element  is  be3'ond  the  ?ith 
letter  of  the  alphabet,  and  no  two  letters  stand  in 
inverse  alphabetical  order,  though  some  may  be 
repeated, 
.*.  each  combination  gives  one  of  the  c^,  with  repetitions  ^  prod- 
ucts ; 

and     •••  all  the  combinations  so  formed  are  unliie, 

.-.    C,(n+r-l)     >    C,,  with  repetitions^- 

.-.    C^,  with  repetitions  ^=    C,(7l -f- r  - 1)  .  Q.  E.  D. 

§  4.     EXAMPLES. 
§2. 

1.  Find  the  number  of  permutations  of  10  things,  all  different, 

taken  3  at  a  time  ;  5  at  a  time  ;  7  at  a  time  ;  all  together. 

2.  Find   the   number  of  permutations  of  10  things,  taken  all 

together,  when  3  are  alike  and  7  alike  ;  when  2  are  alike, 
3  alike,  and  5  alike. 

3.  In  how  many  different  waj^s  can  the  letters  of  the  continued 

product  a^ 6^  be  written?  ofa"6V?  of  ab^c^d^e'^? 

4.  How  man}'  permutations    can   be   formed  from   the  word 

Cornell  ?    Washington  ?   Constantinople  ? 


6.  §4.]  EXAjNIPLES.  119 

5.  In  how  mail}'  wa3^s  can  8  men  stand  in  a  row  ?   12  men  ? 

16  men  ?  n  men  ? 

6.  In   how  man}^  wa3's  can  8  men  sit  at  a  round  table  ?     12 

men  ?   1 6  men  ?  w  men  ? 

7.  Of  how  many  things,  all  different,  are  there  720  permuta- 

tions ?  of  how  many,  all  different  and  taken  3  at  a  time, 
are  there  210  permutations? 

8.  How  many  different  permutations,  taken  three  at  a  time, 

can  be  formed  from  the  word  science  1  from  the  word  con- 
■  stitution  9 

9.  Write  out  the  several  permutations  and  combinations  of  the 

4  digits  1,  2,  3,  4,  taken  1  at  a  time,  2  at  a  time,  3  at  a 
time,  4  at  a  time. 

10.  Find  all  the  factors,  prime  and  composite,  of  6  ;   of  30 ; 

of  240;  of  2310;  of  30030;  ofa6;  of  a6c  ;  of  a^^c^;  of 

abed ;  of  a"*  —  x*. 

§3. 

11.  Find  the  number  of  combinations  of  10  things,  all  different, 

taken  3  at  a  time  ;  5  at  a  time  ;  7  at  a  time.  Show  from 
the  example  why  the  number  of  sets,  taken  3  at  a  time, 
is  the  same  as  the  number  taken  7  at  a  time,  and  why 
the  number  taken  5  at  a  time  is  largest  of  all. 

12.  How  many  triangles  can  be  formed  by  joining  3  vertices 

of  a  polygon  of  n  sides?  how  many  quadrilaterals  by 
joining  4  vertices?  how  many  pentagons  by  joining  5 
vertices  ? 

13.  If  a  line  be   cut  at  4   points,  how  many  segments   are 

formed  ?  at  6  points  ?  at  9  points  ?  at  n  points  ? 

14.  If  there  be  4  straight  lines  in  a  plane,  whereof  no  two  are 

parallel,  and  no  three  meet  in  a  common  point,  how 
many  triangles  are  formed?  if  5  lines?  if  8  lines?  if  n 
lines? 

15.  In  how  many  ways  can  10  things  be  made  up  into  a  set  of 

2,  a  set  of  3,  and  a  set  of  5  ? 

16.  How  many  different  sums  of  money  can  be  formed  from 

1  cent,  1  half-dime,  1  dime,  3  quarter-dollars,  5  dollars? 


120  PERMUTATIONS   AND   COMBINATIONS.         [IV.  §  4. 

17.  From  a  part}'  of  6  ladies   and  7  gentlemen,  how  many 

companies  of  4  ladies  and  4  gentlemen  can  be  formed? 
how  man}*  sets  of  4  couples  for  a  dance  ? 

18.  If  the  number  of  combinations  of  n  things,  taken  4  at  a 

time,  be  twice  the  number  of  permutations,  taken  3  at 
a  time,  what  is  the  number  7i? 

19.  Of  the  combinations  of  8  letters,  "a,  6,  c,  •••,  taken  4  at  a 

time,  how  many  contain  both  a  and  b?  a  and  not  b  ? 
neither  a  nor  b  ? 

•  ••  25.    Show  that: 

20.  p,(7i-f  l)  =  Pr»^4-r-P,_i(n-l)+r.(r-l).p,_2(?i-2)+... 

+  r!-Po(>i  — r  +  1) 
=  p^7i  +  Pir-p^_i(/i  —\)-}-P2r'Pr-2{n  —  2)  +  '" 
-f-p^r.Po(w- r-f-l). 

21.  Pr(m  -h  7l)  =  P^??l4-Pl^-Pr-l^"  Pl^  +  Pi'^  •  Pr-2^'^  *  P2^  +  *•• 

22.  c,(n+l)=c,_in  +  c,_i(7i-l)4-c,_i(7i-2)+- 

+  c,.i(r-l). 

23.  c^(?7i4-n)  =  c^m  +  c^_im«Cin4-c^_2m-C2W  +  ••• 

-f  C'l  ??l  •  C^_i  71  +  c^  w. 

24.  c^(m+7i+jpH — )  =  c^m-|-c^_im-CinH-c^_im-CipH 

+  c^_2Wi-C2n  +  c^_2m.C2pH 

+  c^_2^^-Ci?i-CipH 

wherein  r,  s,  ^,  •••  are  an}^  numbers  such  that 

r  -^  s  -^  t  -\-  '"  —  m-\-n  +  p-j-  '". 

25.  Of  n  things,  all  different,  taken  r  at  a  time,  repetitions 

allowed,  there  are  n*"  permutations. 

26.  Discuss  the  general  case  P^^pauke,  g alike,...' 

27.  Discuss  the  general  case  c^w, alike, « alike,...* 


V.  §  1.]  PRODUCT   OF  BINOMIAL  FACTORS.      *  121 

V.     POWERS  AND  ROOTS  OF  POLYNOMIALS. 

§1.     PRODUCT  OF  BliS^OMIAL   FACTORS. 

Lemma.     If  there  he  n  binomial  factors  (x+a),  (x  +  b),  ••• 

(x+1),  their  product  is  x°-f-2i(a--.l)-x"-^H- 5o(a...l)-x"-2H 

+  2r(a-..l).x"-'H |-2n(a...l),  wherein  2i(a...l),  2o(a.^.  1), 

•  •  •  =  the  sums  of  the  products  of  the  combinations  of  the  terms 
a--- 1,  taken  one  at  a  time^  taken  two  at  a  time^  etc.    [IV.  pr.  2  nt. 

For  • .  •  the  product  {x  +  a)  •  {x  +  6)  •  •  •  (a;  +  Z)  is  the  sum  of  the 
partial  products  of  each  term  of  the  binomial  factor 
{x-\-a)  by  each  term  of  the  binomial  factor  {x-\-h) 
by...,  [ILth.5 

.*.  that  product  =  a;",  the  single  product  of  the  first  terms 
of  all  the  n  binomials, 

+  2i(a ... Z) . aj'*"^,      the  sum  of  the 
n  partial  products  formed,  each  of  them,  b^^  taking 
the  second  term  of  one  binomial,  and  multipl3'ing 
by  the  first  terms  of  all  the  other  n  —  1  binomials, 
-}-  So  (a .  • .  /) .  a;"-2,     the  sum  of  the 
CgW  partial  products  formed  b}'  taking  all  possible  com- 
binations, two  at  a  time,  of  the  second  terms  of  the 
several   binomials,   multiplying  these   two   terms 
together,  and  multipl3'ing  each  such  product  by 
the  first  terms  of  all  the  other  n— 2  binomials, 
+ , 

+  :Sr  (« •  •  •  0  •  ^''"''5     t^®  s"^  ^^  *^® 

C^n  partial  products  formed  by  taking  all  possible  com- 
binations, r  at  a  time,  of  the  second  terms  of  the 
several  binomials,  multiplying  these  terms  together, 
and  multipljing  each  such  product  by  the  first  terms 
of  all  the  other  n  —  r  binomials, 

+ , 

4-  a .  6  ...  ?,  the  single  product 

of  the  last  terms  of  all  the  n  binomials.         Q.  e.  d. 


122  POWERS  AND   ROOTS   OF  POLYNOMIALS.         [V.  th. 


§2.     THE    BINOMIAL   THEOREM. 

Theor.  1.  If  a  binomial  he  raised  to  any  positive  integral 
power ^  that  power  consists  of  the  sum  ofasenes,  whose  successive 
tei-ms  are  the  products  of  three  factors: 

1.  The  powers  of  the  first  term  of  the  binomial,  beginning  with 
that  power  whose  exponent  is  the  exponent  of  the  binomial,  and 
decreasing  by  a  unit  for  each  term  to  the  0th  power. 

2.  The  powers  of  the  second  term  of  the  binomial,  beginning 
with  the  Ofh  power,  and  increasing  by  a  unit  for  each  term. 

3.  The  number  of  combinations  of  a  number  of  things  equal  to 
ike  number  of  units  in  the  exponent,  taken  0,  1,  2,  3,  •••  ai  a  time. 

Let  x-\-a  be  an}'  binomial,  and  n  an}'  positive  integer,  then  will 

1]  {x-^ay=x--^-na^-'-h ^K^-l)^2^n-2^^0^-l)  (»-2)^8^n-8 

+  ...-!_  n{n-l)(n-2)-.-{n-r+l)  ^,^,_, 
r\ 

For,  in  the  equation 

(x-{-a)-{x  +  b)'"{x  +  l) 

=a;"+2i(a...0-a;""^4-22(a...Z).a5"-2-f...  [lemma 
put  a  for  each  of  the  numbers  b,  c,  "•  I; 
then    •.•   (x-{-a)'{x-\-b)"-(x  +  l)  =  {x-\-a)'{x-^a)"'nfsLCtoYS 

=  (a;  +  a)^ 
and     •.•  2i(a---Z)     =a  +a +a  •••  n  terms         =Cin.a, 
22(a  •  •  •  Z)     =  a^  -\-a^-\-a^ '"  C2?i  terms     =  C2^ •  a^ 

2^(a  "•  I)     =  a*"  4-  a**  +  a*"  •  •  •  c^ri  terms     =  c^n  •  a*", 

2„_i(a-"  0=<^'*"^+«""^  H c„_in  terms  =  c,i_in-a"-\ 

2»(«  •  •  •  0    =  a"  once  =  c„n  •  a" ; 

.*.  (a;+a)*=Co?i  •  a"  •  x^'+Cin  -  a  •  ic^-^+CsW  •  a^  •  x""-^ 

H hc,n.a'-.a;^-'-H hc^^in-a^-'^-x+c^n-a'^'aP, 


(n-l) 


i.e.,  (a;  +  a) "  =  a;"  +  nax""-^  +  -^^^ — tl a^ a;n-2  _|. . . . 

7*  ! 

Q.E.D. 


1.  §2.] 


THE  BINOMIAL  TIIEOEEM. 


123 


Note  1 .    The  theorem  is  also  proved  bj-  aid  of  [IV.  th.  2] . 
For  (x  +  ay  =  (x -^ a) ' {x -\-  a) •  {x  +  a)  "•  n  factors 


X'X'X'"X-{-X'X'X 


-\-X'X'a"'X 
-\-X'a'X-"X 
-\-a'X'X'"X 


H \-a-a'a' 


-\-X'X'X"-a'a 

+ 

-}-x-X'a"'X'a 
^x-a-X'-'X-a 

+ 

-\-a»a-X"-x-x 

=  P«Wnalike-a5"  +  P«^«-lalike-«-a;""^  +  P„W«_2alike,2alike-0^^-aj''"^ 
-\ f-  P„W  „_r  alike,  r  alike  '  «*"a;~-''  -\ f-  Pn^nalike  '  »" 

=  x*"  +y^«a;""^  _^n{n  —  1) ^2^n-2 ^  . . .  ^  ^n^   q.e.d. 

Note  2.     The  theorem  is  also  proved  by  induction. 

1 .  TJie  law  is  true  for  the  second  power. 

For     •.*   (a;  +  a)^  =  a^  +  2aa;  +  a^  [multiplication 

.*.    (a;  +  a)"=a;''  +  waic*'~^H f-a"?     whenw=2. 

2.  If  the  law  he  true  for  the  "kth  power,  it  is  also  true  for 
the  {k  +  l)th. 

For,  write  (x  +  a)*=  »*  +  kax^-^  +  Tc{k-1)  ^2^-2^  ... 

A:(fe-l).-.(fe-7t+l) 
A! 
Multiply  both  members  by  a;  +  a  ; 


.ot»/c*"*H |-ct*-      [h}l). 


then 


{x  +  ay+^=a^+^-\-k 
+  1 


^       2! 


a2a^-i+... 


A;(A;-l)...(fc-A4-l) 


^! 


A;(A;-l)...(fe-7^4-2) 
(7.-1)1 


a*a^ 


-A  +  l 


+  ...+a' 


*+i 


2! 


For 
So 


7i! 

77^6  7aw  is  true^  whatever  the  exponent  k. 
it  is  true  for  7i;  =  2, 
it  is  true  for  k=S. 
for  Zu  =  4,  for  A;  =  5,  •  •  •  for  A;  =  n. 


[1 

[2 


Q.E.D. 


124  POWERS  AND   ROOTS   OF  P0LYX0:MIALS.         [V.  th. 

Cor.  1.  If  s.  and  a  he  any  numbers  and  n  a7\y  positive  integer^ 
2]  (x-a)°=x°-nax°-^+  "^^  ~  ^\a^x°-^ q:nax°-i±a°. 

Cor.  2.     The  series  is  finite. 
For     •.•  the  series  is  a  continued  product  of  finite  polynomials, 
.'.  it  is  itself  finite  q.e.d. 

Note.  Another  and  independent  proof  of  Cor.  2  is  as  follows : 
For    •••  the  several  coefficients  form  a  series 

-      n      n     n— 1      n     n—1     n—2 
'     T'     1  *      2    '     1  *      2     '     3    '  *"' 
wherein  each  term  is  formed  by  multiplying  the  preceding  one  by 

a  fraction  of  the  form     ~     ;  [IV.  th.3  nt. 

Aj  -p  1 

and    •.•  the  numerator  of  this  fraction  grows  less  by  a  unit  at 

each  step,  and  the  denominator  greater, 

.*.  some  term  of  the  series,  and  all  after  it,  is  0,  and  the 

series  terminates. 

Cob.  3.  The  coefficients  of  any  two  terms  equally  distant  from 
the  extremities  of  the  development  are  identical. 

CoR.  4.     The  sum  of  the  coefficients  of  (x  +  a)°  is  2°. 
For,  let  a;=  1,  a=  1 ; 
then   •.•   (a;  +  a)"  =  (14-1)"  =  2", 

and    •.•   (l  +  l)'*=l»+7i-l.r-^+^^^^~'^^-l'-l""'+ ••♦+!" 
=  l  +  .  +  !fci)4....  +  l, 

...    2»  =l  +  n+^^(!^=i)+...  +  l.  Q.E.D. 

CoR.  5.    The  sum  of  the  coefficients  of  (x  —  a)'^  is  0. 

CoR.  6.  In  the  development  of  (x  +  a)°  the  sum  of  1st,  Bd, 
5th,  •••  coefficients,  and  the  sum  of  the  2d,  Uh,  6th,  •••  coefficients, 
are  equal;  and  each  sum  is  2^~^. 

For     •.•  the  sum  of  aU  of  them  is  2",  [cr.4 

and     •.•  sum(lst+3d+-..)  — sum(2d+4th+..-)  =  0,         [cr.  5 

.-.  sum  (lst+3d-f ...)  =  sum  (2d4-4th+...)  =  2*» :  2  =  2""^ 


2.  §  3.J         THE  POLYNOMIAL  THEOREM.  125 

Note.    Cors.  4,  5,  6  may  be  written  in  formula,  thus : 

3]  c„n  -f  c„_i7i  -f  c„_2W  H h  02^  +  CiTi  =  2'», 

4]  c„?i  —  c,,_in  -\-  c„_2/i :f  C2?i  ±  Cin  =  0, 

5]  c„7i  +  c„_2n  +  c„.47i  +...=  c„_i7i+c„_3n4-o„_5?i=2"-i. 

§3.     THE   POLYNOMIAL   THEOEEM. 

Theor.  2.  7/*  a,  b,  c,  •••  1  6e  a?i?/  m  numbers;  n  a  positive 
integer;  p,  q,  r,  •••  z  any  positive  integers  (including  0),  such  that 
p  +  q  +  rH |-z  =  n,  ^/ien; 

6]  {a  +  h  +  c  +  -"+lY 

n\ 


n!.0!.0! 
n\ 


(71-1)!. 1!.0! 


4- 7 ^.?\.   ^. :Sa"-2-62.c°...Z« 


(w-2)!.2!.0!... 

n\ 

(7i  —  2) ! .  1  ! .  1  ! .  0  ! 


:Sa"-2.6i.c^.d*'...?' 


'^^  .^a^'-^-J/'C^'-'l^ 


(7i-3)!.3!.0!... 

H — ^a^'-^'h^-G^'dP'-'l^ 

(7i-3)!.2!.l!.0!... 

+7 rr-; — ^-; iSa^'^  •  ft^  •  c^ .  d^ .  e«  .••  ?> 

(n-3)!.l!.l!.l  !.0!..- 

+ 

n ' 

H — ; ; — '- %a^'¥'C''"'l'   [the  general  term 

p\'q\'r\-"Z\ 

+ 

This  theorem  is  but  the  generalization  of  the  binomial  theorem, 
and  is  proved  in  the  same  wa}^ 

The  reader  ma}^  review  here  what  is  said  of  sj^mmetry  in 
multiplication  [II.  pr.  3 ,  nt.  7] .  He  may  also  compare  [IV.  th.  5] . 
He  will  observe  that  he  is  actually  forming  the  homogeneous 
products  there  spoken  of.  They  are,  however,  of  the  wth  degree 
here,  instead  of  the  A;th  degree  as  there,  and  there  are  m  numbers 
instead  of  n. 


126 


POWERS  AND  EOOTS   OF  POLYNOISIIALS.         [V.  pr. 


Cor.  1.     Let  a-f-bx-f  cx'+dx^H he  a  senes  arranged  to 

ascending  powers  ofx;  then  will 
7]  (a  +  bx  +  cx^  +  dx3  +...)" 


n!»a°       n!.a°"^b 
n!     "^(n-l)!!! 


x-f- 


n!-a" 


'b^ 


(n-2)!2 


(n-l)!l 


xH 


n!.a°-^b^ 
(n-3)!  3! 


n ! .  a^ 


be 


(n-2)!l!l! 
n!.a°-M 
(n-1)!  1! 


x«+- 


=  a°+na°-^bx-f^(^Y^^^°"'^' 


H-na' 


n-l. 


.2  .  n(n-l)(n 
3! 


-''a"-«l 


x^+ 


4-n(ii-l)a"-2bc 

+  na°-M 

and,  if  p  be  any  positive  integer,  and  r,  s,  t  •••  be  any  other 
positive  integers,  such  that  0 •  r  +  1  •  s  +  2  •  t  +  •••  =  p,  the  co- 
efficient ofsJ^in  the  development  is    %■ 


nl 


,    .      .      ,      .a'.b'.c*.... 
r!.s!-t!-.. 

Cor.  2.  If  all  the  m  numbers  a,  b,  c,  •••  1  6e  positive,  the 
sum  of  the  coefficients  of  the  development  o/  (a  +  b  +  c  +  •••1)'^ 
is  m^  ;  if  one  of  them  he  negative,  the  sum  is  (m  —  2)°  ;  if  two  of 
them  be  negative,  the  sum  is  (m  —  4)°,  •••  ;    and  so  on. 

Cor.  3.  The  development  has  as  many  sums  of  symmetric  terms 
of  the  form  given  above  as  there  are  ways  in  which  m  positive  in- 
tegers p,  q,  r,  •••  z  can  be  chosen,  so  that  their  sum  shall  be  n. 

E.g.,  if  m  =  4,  and  n  =  6,  the  four  integers  p,  q,  r,  s  may 
be  either  of  the  following 


6,  0,  0,  0 
4,1,1,0 
3,  1,1,1 


5,  1,  0,  0 
3,  3,  0,  0 

2,  2,  2,  0 


4,  2,  0,  0 
3,2,1,0 

2,  2,  1,  1 


and  there  are  nine  terms  in  the  development. 


Cor,  4.     The  development  has 


(n  +  m-l)I 


separate  terms. 


n!.(m-l)! 

For  this  is  the  greatest  number  of  terms  possible  in  any  integral 
pol3'nomial  of  the  nth.  degree  homogeneous  and  having  m  letters. 

E.g.,  (a+6a;+ca^)^has  ,  =15,  separate  terms.  [IV.  th.6 


1.  §  4.]  BOOTS   OF  POLYNOMIALS.  127 

§  4.     BOOTS    OF    POLYNOMIALS. 
PrOB.   1.      To   FIND   THE    nth   ROOT   OF   A   POLYNOMIAL. 

Arrange  the  terms  of  the  polynomial  in  the  order  of  the  powers 
of  some  one  letter,  a  perfect  power  first. 

If  the  first  term  be  not  a  perfect  power,  divide  the  polynomial 
by  such  a  monomial  as  will  make  it  a  perfect  power,  and  reserve 
the  root  of  this  monomial  as  a  factor  of  the  result. 

Take  the  nth  root  of  the  first  term. 

Raise  this  root  to  the  {\\  —  \)th  power  and  multiply  by  n. 

Divide  the  second  term  of  the  polynomial  by  this  product  (the 
trial  divisor)  and  add  the  quotient  to  the  root  first  found. 

Raise  the  whole  root  to  the  nth  power  and  subtract  it  from  the 
polynomial. 

Divide  the  first  term  of  the  remainder  by  the  trial  divisor;  add 
the  quotient  to  the  root  found;  raise  the  whole  root  to  the  nth 
power;  subtract  from  the  polynomial ;  and  so  on. 

Let  p  =  the  given  polynomial,  and  a  H d  +  e  H =  its  nth 

root,  both  arranged  bv  -{    ^^,^^^y^^^  powers  of  some  letter  x; 

and  let  a  H d  =  the  terms  already  found  ; 

then   •.•  p  — aH d"  =  (aH d  +  eH )"  — aH d" 

lower 
=  WA""^  •  E  -|-  terms  with  -{  ,.,       powers 

of  X,  "° 

.  • .  E  =  first  term  of  quotient,  (p  —  a  H d**)  :  tia**"^, 


and  p  — A  H d  4-e  , 

„_i    _    ,   ^ __,,,.    ,  lower 


=  71A' 


F  +  terms  with  ^  u-q-i,      powers  of  x, 


highest 


has  not  the  ^i  i  ^     1    power  of  a;  in  p  —  a  -|-  •••  d  . 

So        the  successive  terras  of  p  are  exhausted,  as  new  terms 

of  the  root  are  found.  q.  e.  d. 

Note  1 .  The  work  is  an  effort  to  retrace  the  steps  taken  in 
getting  the  power  whose  root  is  now  sought.  It  is  a  process  of 
trial,  by  progressive  steps,  like  division  and  other  inverse  opera- 
tions, and  its  success  is  established  by  raising  the  root  to  the 
required  power  and  comparing  it  with  the  given  polynomial. 

[II.  §  2,  p.  29 


128  POWERS   AND   ROOTS   OF  POLYNOMIALS.         [V.  pr. 

Note  2.  Complete  Divisor  :  In  square  root  and  cube  root 
certain  modifications  may  be  introduced  into  the  rule  which 
shorten  the  work : 

In  square  root  the  trial  divisor  is  double  the  first  term  of  the 
root ;  and  a  complete  divisor  is  got  by  doubling  the  root  already 
found  and  adding  the  new  term  of  the  root,  \yhen  the  complete 
divisor  is  multiplied  b}"  this  new  term  of  the  root,  and  the  prod- 
uct is  subtracted  from  the  last  remainder,  the  whole  root  found 
is  thereby  squared  and  subtracted  Irom  the  polj'uomial. 

E.g.,  a^  +  2ab  -hb^'  -^2ac  -\-  2bc  -i-  c^  \a-\-b  +  c 

a  > 


2a-j-b\__2ab_±J/ 

2a4-2b4-c  I  2ac  +  2&c  +  c^ 
In  cube  root  the  trial  divisor  is  three  times  the  square  of  the 
first  term  of  the  root,  and  the  complete  divisor  is  the  sum  of 
three  times  the  square  of  the  root  already  found,  three  times  the 
product  of  this  root  b}'  the  new  term  of  the  root,  and  the  square 
of  the  new  term  ;  and  when  the  complete  divisor  is  multiplied  by 
the  new  term  and  subtracted  from  the  last  remainder,  the  whole 
root  found  is  thereby  cubed  and  subtracted  from  the  polynomial. 
E.g.,    |q  +  fc  +  c 

cr*  +  3  a'-*  6  +  3  a6H  &H  3  a' c  +  6  a6c  +  3  ac2+ 3  ft'-'c  +  3  6c'-*+ 0^ 

o^ 

3a^+3ab  +  b^\  Sa'b  +  Sab''-^b^ 

3a^+6«&  +  36H3ac4-36c  +  c'-'  |  3a'^c  +  6a6c  +  3acH3  6'-^c+36c^+c* 

The  reader  may  deduce  like  rules  for  getting  the  4th,  5th,  ••• 
roots,  by  means  of  the  complete  divisor,  from  the  formula 

A«  +  (71A"-1+  '-ii^^V-2B  +  ...  +  B'^-l)  .  B  =  (A  +  B)^  [1 

Note  3.  Roots  of  Roots  :  For  a  root  whose  index  is  composite, 
it  is  generally  bettor  to  factor  the  index  and  take  in  succession 
the  roots  indicated  b}-  such  factors.  [II.  th.  3  cr.  9,  nt. 

E.g.,    the  4th  root  is  the  square  root  of  the  square  root ; 
the  6th  root  is  the  cube     root  of  the  square  root ; 
the  8th  root  is  the  square  root  of  the  square  root  of  the 
square  root ;    and  so  on. 
Note  4.    Roots  of  Fractions  :  To  find  the  root  of  a  fraction, 
write  the  root  of  the  numerator  over  that  of  the  denominator. 


1.  §  5.]  ABSOLUTE  AND   RELATIVE  EREOE.  129 


§5.     ABSOLUTE    AND    RELATIVE    ERROR. 

When  a  number  is  given  approximate!}^  only,  the  absolute 
error  is  the  excess  of  the  assumed  value  above  the  true  value ; 

and  it  is  -(  P       , .     if  the  assumed  value  be  -i  f  than  the  true 

'  negative  '  less 

value.    The  relative  error  is  the  ratio,  absolute  error :  true  value. 

The  possible  error ^  whether  absolute  or  relative,  is  the  smallest 

number  than  which  the  actual  error  is  known  not  to  be  larger. 

o 

E.g.^  if  of  a  long  decimal  a  few  figures  only  be  given,  the  last 
figure  written  is  usually  increased  by  1  when  the  first  figure 
dropped  is  5  or  more ;  and  the  possible  error  is  then  only  half 
a  unit  of  the  last  place  written. 
correct 

A  number  is  {  approximate  ^^  ^fi^^'^'^^  ^^^^  i*s  absolute  error 

is  not  larger  than  ■{  ^  unit  in  its  nth  place  towards  the  right. 

E.g.,  if  ic—  .2037  >  .0005,  then  .204  is  approximate  to  three 
figures,  and  .20  is  correct  to  two  figures. 

So,  for  100a;,  20.4  is  approximate  to  three  figures. 

The  copula  = ,  read  approaches,  joins  numbers  which  diflfer  by 
a  number  ver^'  small  as  to  either  of  them.  It  is,  therefore,  used 
to  join  an  assumed  value  to  the  true  value  of  a  number  when  the 
relative  error  becomes  very  small. 

E.g.,  if  a  be  the  true  value  of  a  number,  x  the  assumed  value, 
and  a  the  absolute  error,  then  x  =  a  -^^  a^  and  a?  =  a  when  a 
becomes  very  small. 

3  3 

So,  3a-fa2  =  3a,  — 1-1=-,    when  a  becomes  very  small. 

In  numerical  work  the  degree  of  approximation  depends  on 
the  relative  error. 

E.g.,  an  inch  in  the  earth's  diameter,  and  a  million  miles  in  a 
star's  distance,  are  alike  inappreciable  ;  but  a  thousandth  of  an 
inch  in  a  microscopic  measurement  is  enormous. 

In  pure  mathematics  the  degree  of  approximation  depends 
solely  upon  the  time  and  patience  of  the  computer  ;  but  of  num- 
bers based  on  measurement  the  positive  relative  error  is  seldom 
smaller  than  a  millionth. 


130  POWERS  AND   EOOTS   OF   POLYNOMIALS.  [Y.  th. 

Theor.  ^.    If  a  number  he  approximate  to  n  significant  figures 
and  no  more,  the  possible  relative  error  >  1  :  10^' and  ^  1  :  lO'^""^ 
For     •••  an}^  number  <  10"  units  of  its  own  nth  place, 
and  ^  10""^  such  units, 

and     *.•  poss. abs. err. :=:  1  such  unit,  [liyp- 

and     •••  poss.  rel.  err.  =  poss.  abs.  err.  :  true  number,  [df. 

.-.  poss.  rel.  err.  >  1 :  10"  and  ^  1 :  10""^  q.e.d.  [ILax.18 

CoK.    A  number  wJtose  relative  error  is  not  larger  than  1 :  10° 
IS  approximate  to  at  least  n  sigyiificant  figures. 
For     *.•  the  number  <  10**  units  of  the  nth  place, 
and     •.*  its  rel.  err.  ^  1 :  10",  [iiyp* 

.-.  its  abs. err.,  =  number  x  rel.  err., 

<  1  unit  of  the  7ith  place  ; 
I.e.,  the  number  is  approximate  to  n  figures.       q.e.d. 

Theoe.4.   Tke^fj^l^^^err.rofthei';;^^^^^oftwoormore 

numbers  -J  ««""'«   ,      the  mm  of  their  {  "^f"'""  errors. 
'  approaches  •'  '  relative 

For,  let  a,  6,  •••  =  the  true  values  of  two  or  more  numbers, 
«,  y,  •  • .  =  their  assumed  values,  a,  /?,  •  •  •  =  their  absolute  errors  ; 
then: 

(a)  •.•  a;  +  2/+...  =  (a  +  a)  +  (6+^)+-- 

=  (a  +  6  +  ...)  +  (a4-^+..-), 

.'.  the  abs.  err.  of  sum,  x  +  y  -\ ,  =a  +  y8  H , 

i.e.,  =  sum  of  abs.  errs.  q.e.d. 

(b)  '.'  a;.2/...=(a  +  a)-(6+y8)... 

=  a'b'-'  +  terms  which  contain  either  a  and 
not  a,  or  (3  and  not  &,  or  •••,  as  a  factor,  +  terms 
with  two  or  more  of  the  abs.  errs,  a,  /3,  •••  as  factors, 

.'.  abs.  err.  prod,  a;- 2/ •••,  ^x-y a-b---, 

=  the  sum  of  terms  all  having 
one  or  more  of  the  abs.  errs,  a,  ;8,  ...  for  factors ; 

.*.  rel.  err.  prod,  x-y  "•  =-  +  ^-| 1 7  -j • 

a      0  a-  0 

i.e. ,  rel.  err.  prod.  a;.2/...=-+^-f-...  =  sum  of  rel.  errs. 

/  Q.E.D. 


3-5.  §5.]  ABSOLUTE  AND  RELATIVE  ERROR.  131 

Cor.  1.     If  the  abs.  errs,  a,  y8,  •••  6e  eacJi  not  larger  tJianc,  and 
(f  m,  n,  •••  6e  any  finite  numbers^  then 
abs.  err.  (mx  +  nyH )  =  ma -f- n/3 -j ^  (+ni  +  "'-nH ).e. 

In  2)cirticular :  abs.  err.  {x  —  y)  =  a  —  l3,  ^  2  e. 

Cor.  2.   Bel.  err.  mx  =  rel.  err.  x,  ifm  have  no  error. 
Cor.  3.    The  relative  error  of  Or  quotient  approaches  the  differ- 
ence of  the  relative  errors  of  the  elements. 
For     •.•  divd.  =  divr.  x  quot., 

.-.  rel.  err.  divd.  =  rel.  err.  divr.  +  I'el.  err.  quot.  ;  [th. 

. • .  rel.  err.  quot.  ==  rel.  err.  divd.  —  rel.  err.  divr.       Q.  e.  d. 

In  particular :  the  relative  error  of  the  reciprocal  of  a  numbei 

approaches  the  opposite  of  the  relative  error  of  the  number  itself. 

Theor.  5.  The  relative  error  of  a  positive  integral  power  of  a 
number  approaches  the  product  of  the  relative  error  of  the  number 
by  the  exponent  of  the  power;  and  that  of  a  root  approaches  the 
quotient  of  the  relative  error  of  the  number  by  the  root-index. 

Let  X  be  any  approximate  number,  and  n  any  positive  integer ; 

then  will  rel.  err.  a;**  ==  n  •  rel.  err.  x.  and  rel.  err.  -^x-^  -  •  rel.  err.  x. 

^        n 

.*.  rel.  err.  a;",  = ,  =  ti  •     +  — ^^ ^ 

a"  ^  2 ! 

=  7i .  -  =  7i .  rel.  err.  x. 
a 

(b)     '.'  x  =  {^xy, 

and     •.•  rel.  err.  x,    =  rel.  err.  (-v/a;)",  =  n-rel.  err.  -y/x  ;    [(a) 

.*.  rel.  err. -?/a;  =  -  •  rel.  err.  a;.  q.e.d. 

^         n 

Note.  Ths.  3-5  enable  the  computer:  (a)  to  see  how  far 
his  results  can  be  depended  upon  as  approximate ;  (b)  to  carry 
each  part  of  his  work  so  far  that  the  final  result  shall  be  as 
approximate  as  he  desires,  or  as  the  data,  if  themselves  only 
approximate,  permit,  wasting  no  labor  upon  needless  or  unreli- 
able figures.  Results  correct  to  the  last  figure,  e.g.  for  standard 
tables,  are  only  got  by  computing  with  several  extra  decimals. 


\aj 

**? 

Q.E.D. 

[I.  §11, 

df. 

132  POWEKS  AND  ROOTS   OF  POLYNOMIALS.  [V.pr. 

§6.     EOOTS    OF  NUMERALS. 
PrOB.  2.       To  FIND  THE  TITH  ROOT  OF  A  NUMERAL. 

Separate  the  numeral  into  periods  of  n  figures  each,  both  to  the 
left  and  to  the  right  of  the  decimal  point. 

Take  the  nth  root  of  the  largest  perfect  nth  power  contained  in 
the  left-hand  penod. 

Subtract  this  power  from  the  period^  and  to  the  remainder 
annex  the  next  period  to  form  the  first  dividend. 

Raise  the  root  figure  to  the  {n  —  \)th  power,  and  multiply  byn. 

Divide  the  first  dividend  by  this  product  (the  trial  divisor) ,  and 
annex  the  quotient-figure  to  the  root  first  found. 

Raise  the  whole  root  to  the  nth  power,  subtract  from  the  first 
two  periods,  and  to  the  remainder  annex  the  next  period  for  the 
second  dividend. 

Raise  the  root  found  to  the  {n—l)th  power,  and  multiply  by  n 
for  a  new  tnal  divisor;   and  so  on. 

Note  1.  Numerals  are  pol3^nomials,  but  pol3'nomials  in  which 
the  terms  overlie  and  hide  each  other ;  and  virtually  the  rule  is 
the  same  for  finding  the  roots  of  both. 

The  separation  into  periods  is  a  matter  of  convenience  only. 
It  comes  from  this  :  that  the  figures  of  the  root  of  different  orders 
are  best  got  separatel}',  and  that,  since  the  nth  power  of  even 
tens  has  n  O's,  therefore  the  first  n  figures,  counting  from  the 
decimal  point  to  the  left,  are  of  no  avail  in  getting  the  tens  of 
the  root,  and  are  set  aside  and  reserved  till  wanted  in  getting 
the  units'  figure.  So  the  nth.  power  of  even  hundreds  has  2n  O's, 
and  the  first  2  n  figures,  two  periods,  are  set  aside  and  reserved 
till  wanted  in  getting  the  tens  ;    and  so  on. 

So,  in  getting  roots  of  decimal  fractions,  the  nth  power  of 
tenths  has  n  decimal  figures,  and  the  first  n  figures,  one  period, 
are  used  in  getting  the  tenths'  figure  of  the  root ;  the  ntli  power 
of  hundredths  has  2n  decimal  figures,  and  so  on.  The  same 
thing  appears  from  this  :  that  the  root  is  easiest  got  if  the  denom- 
inator be  a  perfect  ?ith  power  ;  and  this  it  is  only  when  it  consists 
of  1  with  n  O's,  or  2n  O's,  or  3?i  O's,  and  so  on  ;  that  is,  when  the 
number  of  decimal  figures  used  is  n,  or  2n,  or  3  n,  and  so  on. 


2.  §6.]  EOOTS   OF  NU]MERAT.S.  133 

Note  2.  Approximation  :  The  root  of  a  numeral  may  be  got 
to  Siuy  degree  of  approximation  by  reducing  it  to  a  fraction  whose 
denominator  equals  or  exceeds  the  ?ith  power  of  the  denominator 
sought,  and  then  extracting  the  root. 

6912  T      3/G912      19  + 


^  '    ^  ''  1728  \l7i 


28        12 

Note  3.    Square  Root  by  Contraction  :   When  the  first  n 
figures  of  the  root  of  a  numeral  have  been  got  bj'  the  rule  above, 
then  71—1  more  figures  may  be  got  by  dividing  the  remainder  by 
double  this  root. 
For    •.•  trial-divisor,   =  2xfirst  7i  figs,  root  ^  2x  10**"^  differs 

from  complete  divisor  by  subsequent  root-figs., 
i.e.,          by  <  1  in  nth.  place  of  root, 
i.e.,  by  <1  :  (2x  10"^)  of  the  whole  divisor, 

.-.  rel.  err.  quotient  <1  :  (2  X  lO'*-^)  ;  [th.  4  cr.  3 

and     *.*  the  quotient  <  10"^^  units  of  its  own  n  —1th  place, 
.*.  abs.  err.  quotient,  in  units  of  its  own  ?i  — 1th  place, 

^lO'^-iCl:  (2x10-0]    =i;  

and    •      this,  with  the  further  poss.  err.  of  ±^  unit  in  w— 1th 

quotient-place  due  to  writing  quotient  no  further  [p. 

129, 1st  e.g."],  gives  total  poss.  err.  <  1  in  that  place  ; 
i.e.,  the  quotient  is  approximate  to  n  —  1  figures, 

and  the  root  is  approximate  to  2w  — 1  figures,     q.e.d. 

Note  4.  Cube  Root  by  Contraction  :  When  the  first  n  fig- 
ures of  a  root  have  been  got  by  the  rule  above,  then  n  —  2  more 
figures  may  be  got  by  dividing  the  remainder  by  three  times  the 
square  of  this  root. 

For     •.•  trial-divisor,     =  3  •  (first  n  figs,  root)^     differs  from 
complete  divisor  by  only  3  -(first  n  figs.  rt.)-(rest 
of  rt.)  +  (restof  rt.)2, 
i.e.,          by  <4 .  first  n  figs,  rt.,         [rest  of  rt.  <1  of  nth.  place 
.'.  rel.  err.  divisor  <  | :  first  n  figs.  rt.  <  | :  10"~S 
.-.  rel.  err.  quotient  <|:  W-\  [th.4cr.3 

and  abs.  err.  quot.,  in  units  of  its  own  n  —2th  place,  <^u<t'^ 

.*.  even  with  the  further  possible  error  of  ±^  in  n  — 2th 
quotient-place  due  to  writing  no  more  figures, 
quot.  is  approx.  to  at  least  n—2  places,    q.e.d. 


134  POWERS  AND  EOOTS   OF  POLYNOMIALS.  [V. 

§  7.     EXAMPLES. 

§  2.      THEOR.  1. 

1.  Expand  (l+a;)«,   (a4-&)^   {^a-2a^y,  (a  +  bx -{-cx'y. 

Note.  To  expand  a  tiinomial,  bracket  two  terms  and  apply 
the  formula  both  to  the  whole  expression  and  to  the  powers  of 
the  bracket,      thus  {a  +  bx -{-cary. 

2.  In  {x  +  yy*,  show  that  the  sum  of  the  coeflScients  of  the  odd 

terms  equals  the  sum  of  the  coefficients  of  the  even  terms. 

3.  Write  down  that  term  of  the  expansion  of  lx-j--\  which 

does  not  contain  x  when  n  is  even.  ^  ^ 

4.  Write  down  the  8th  term,  and  the  largest  term,  of  ( 1  +  -  ]  • 

Note.  To  determine  the  largest  term,  observe  the  factors  by 
which  the  successive  terms  are  multiplied  to  get  the  next  terms 
in  order.  These  multipliers  constantl}^  grow  smaller  ;  and  when 
first  one  of  them  falls  below  a  unit,  then  the  term  last  before  it 
is  the  largest,  and  those  which  follow  are  successively  smaller 
and  smaller.  Sometimes  two  successive  terms,  equally  large, 
are  larger  than  any  of  the  others. 

5.  By  means  of  the  binomial  theorem  show  that  the  number  of 

all  possible  combinations  of  8  things  is  255. 

6.  Show  that  the  coefficient  of  the  9th  term  in  the  expansion 

of  (1 4-^)^  is  equal  to  the  sum  of  the  coefficients  of  the 
8th  and  the  9th  term  of  the  expansion  of  (1  +  xy^. 

7.  In  Ex.  6  write  n  in  place  of  11  and  r  in  place  of  8,  and  make 

the  proof  general. 

8.  Show  that  the  middle  term  of  the  expansion  of  (1  +  a;)^"  is 

l-3.5...(2n-l).2'».a;»:?i! 

9.  Kthe  coefficients  of  the  (r+l)th  and  the  (r-|-3)th  terms  of 

the  expansion  of  (1-f-  «)**  be  equal,  find  r. 
10.    If  N  =  the  ?ith  term  of  the  expansion  of  {l  —  x^^  then  the 
series,  after  the  first  n  tei-ms,  is 


.,.(,_=±i)„.^.(,_-±.).(,-=±i) 


+  •• •. 


§  7.]  EXAMPLES.  135 

§  3.     THEOR.  2. 

1 1 .  Write  seven  terms  of  the  expansion  of  (a-^bx-{- cx^-\ —  )*. 

12.  Write  the  expansion  of  (l  —  5x-\-Sx^y. 

1 3 .  Write  eight  terms  of  the  expansion ,  ( 1  —  cc — a^ + r^ + a;^ y. 

14.  Write  the  expansion  of  (a4- Ba;^+  co;^  +  •••)^  as  far  as  x^^. 

15.  Expand  (a -\-b -{-c-\- dy,   (a +  6 +  c +  d +  e)^  in  sums  of 

S3'mmetric  terms.  • 

How  many  unlike  terms  in  each  of  these  sums  ? 
How  man}'  partial  products  in  each  sum? 
Check  the  work  by  showing  that  the  number  of  unlike  terms 
in  all  the  sums  is  as  in  [IV.  th.  6] ,  and  that  the  number 
of  partial  products  in  all  the  sums  is  as  in  [II.  th.  5  cr.  7]. 

§  4.     PROB.  1. 

16.  Extract  the  square  roots  of: 

IQsc^-  4.0xy  +  25^2 .     i  _^2x-{-7x' -{-Gx^ +  dx* ; 

dx'-n0ax-3a'x+2oa'+5a'+^;   -^+t^-+i-l. 

4      y-     or    y    X    4: 

1  4-  a^,    a;^  +  1 ,    xr  —  a^,   and  a^  —  x^,  each  to  4  terms. 

17.  Extract  the  cube  roots  of: 

l+6a;+12a;2_^g3,3.   a«- 9 a^ft'^c  +  27 a^^^c-- 276V. 
1  -  6a; -I- 21  a^  -  44 a:^  +  63a;*  -  o4x^  +  27a;« ; 

^-Qx^-^-Ux'f-Sy^;     ^4-1+    2    +_^; 
f  ^         ^  '      8      2      3a;«      27a;« 

{a-\-iy''af-6ca%a+iy''x^+12(fa^p(a+iy''x-8<^a^', 

18.  Extract  the  4th  root  of 

x*-\-  4ar^7/-|-  6ar?/2+  4xf-\- y^+4:X^z  +  12x^yz  +  Uxy^z 
+  4fz  +  6ar^22  ^  Uxyz"  J^Qy'-z'-\-4.xz^  +  ^V^ -^A 
i.e.,  of  •^x*-{-%4ii(^y  +  ^ea^y^  +  ^V2xFyz. 

19.  Extract  the  5th  root  of 

x^^-^x^y+^x^y^-^a^f-\-^x'y'--^f. 
2     -^^2  4  16  32"^ 

20.  Extract  the  6th  root  of  [at  one  operation,  or  at  two 

a^-2a'b  +  -a'b'-—a'b'-[-~a'b'-^ab'-{-^b\. 
3  27  27  81  729 


136  POWERS   AND   EOOTS   OF   POLYNOMIALS.       [V.  §  7. 

§  G.     PROB.  2. 

21.  Extract  the  square  root,  each  to  three  decimal  places,  of 

144,    14.4,   1.44,    .144,    .0144,    .00144,    .000144. 

22.  Extract  the  cube  root,  each  to  three  decimal  places,  of 

1728,   172.8,   17.28,   1.728,   .1728,   .01728,   .001728. 

23.  Find  the  values,  each  correct  to  within  ^,  of 

.._       117.     (17       117       181       |81       /81       181 
^''"\9'   W   W   \12'   W  \63'   A|324* 

24.  Find  the  values,  each  correct  to  within  |^,  of 

3/xo     3/53     8|53     8p3     J53     J53     J  53      7/53 
^^^'  \12'  \T'  \27'   \T  \36'  \^6    \729* 

25.  Find  the  values,  each  to  five  decimal  places,  of  [contraction 

V185,  V912,  -^625,  ^587,  </729,  -^1008,  ^1728. 


26.  Jfx'*^  =  X'{x-\-d)'{x-^2d)--'(x-^n—l'd)   [71  a pos. integer 

show  that    af '®  =  xT,     a;^'**  =  a;,     a/^-'*  =  1 . 

27.  If  p  =  an}'  homogeneous  polynomial  of  the  nth  degree  as  to 

a;, 2/,  •••,  and  if  q,  R  =  whatp,  (x-\-y-\ )-p  become 

when  for  ar^,  a^,  •••  y^  •••  are  put  a^'**,  x^''^^  •••  ?/^''',  •••, 
show  that  every  term  aaf''^y'''^  •••  of  q  gives  in  the  product 
{n-\-x-j-y-\--'-)'Q  the  terms  ax""^^^^ y'^^  •••,  aaf''^y'+^^^  .••, 
etc.  ;  and  hence,  that  (n-{-x-\-y  -\ ) .  q  =  r. 

28.  Use  the  result  of  Ex.  27  to  show  that : 

(a;+2/)"'''  =  ^"''*  +  n •  x^'^^'' ■  y  +  ^^^~^^  •  x'^-'^"  - /•'* 


n\ 


\d 


plqlrl'-' 

[p-\-q  +  r+ 


VI.  §  1.]  FORM  OF   CONTINUED  FRACTIONS.  137 

VI.     CONTINUED   FRACTIONS. 

§  1.      FORM   OF   CONTINUED   FRACTIONS.  —  CONVERGENTS. 

In  place  of  fractions  with  large  terms,  or  of  incommensurables, 
it  is  often  convenient  to  use  fractions  with  small  terms,  whose 
values  are  nearly  equal  to  the  true  values  of  the  given  numbers. 
Such  approximate  fractions,  when  arranged  in  a  series  approach- 
ing more  and  more  closel}^  to  the  true  value  of  a  number,  are 
called  its  convergents.  The  excess  of  a  convergent  over  the 
true  value  is  its  error.  Convergents  are  found  in  various  waj^s  ; 
among  others,  by  aid  of  continued  fractions. 

A  continued  fraction  is  an  expression  of  the  form 

"i  +  -, — :  n. 


7l2 

d^+ 


"3 


c?3  4--.    ,  n^ 


I.e.,  a  complex  fraction  whose  numerator  is  an  entire  number, 
and  whose  denominator  is  an  entire  number  plus  a  fraction  whose 
numerator  is  an  entire  number  and  whose  denominator  is  an 
entire  number  plus,  and  so  on. 

The  fractions  ^,   -S    %  •••  -*  are  the  1st,  2d,  3d,  ...  A:th 
di    d^     d^         djf 

partial  fractions,  and  ■{   ,^'    ,^'  ^^'  "*    ,*  are  the  1st,  2d,  3d,  ••• 
«!,  a2,  a3,  ••.  «j 

A:th  partial  -{   .  .     .  '        These  partial  numerators  and  de- 

nominators are  here  assumed  to  be  entire  numbers,  and  they 
may  be  either  positive  or  negative. 
The  expressions 

Til .  ^1  n^d^     ,       ni  _       . 

a,     ch+-       d,d,  +  n,         ^1  +  ^.^       ^^ 

are  called  the  1st,  2d,  ...  kth.  convergents,  because,  usually, 
they  are  true  convergents  ;  but  sometimes  they  do  not  converge 
toward  the  true  value,  but  diverge  from  it. 


138  CONTINUED  FRACTIONS.  [VI.  pr. 

A  continued  fraction  is  -j  "^^l   ..    when  the  number  of  partial 
fractions,  and  therefore  of  convergents,  is  ^      ,.    ./  , 

An  infinite  continued  fraction  -{  ^^.^'^^^5'^^   when  its  conver- 

'  diverges 

gents,  if  carried  far  enough,  -{  ^L       ,  differ  from  the  true  value 
by  less  than  any  assigned  number,  however  small. 


§  2.    CONVERSION   OF   COMMON   FRACTIONS. 

PrOB.  1.  To  CONVERT  A  COMMON  FRACTION  INTO  A  CONTINUED 
FRACTION. 

(a)  Numerical,  ni,  ng,  •  •  •  each  =  1 : 

If  an  improper  fraction,  reduce  to  a  mixed  number;  then  di- 
vide both  teims  of  the  fractional  part  by  its  numerator,  both  terms 
of  the  fractional  part  of  the  new  denominator  by  its  numerator, 
and  so  on. 

'^''     248      3H     3H-i—      3  +  ^1        3  +  i-i 


and  its  convergents  are : 


2  -Tg? 


1.1_        ^1_.     h.  =^.1_  ^   79 

3'     3  +  -'      22'     3+—  l'       113'     3+—  i  '       248' 

whereof  the  last  is  the  original  fraction. 

Note.  The  reader  may  find  the  h.  c.  msr.  of  79  and  248.  He 
will  observe  that  the  divisions  made  above  in  converting  the 
common  fraction  into  a  continued  fraction  are  precisely  the  same 
as  those  made  in  getting  the  h.  c.  msr.  of  the  numerator  and  de- 
nominator, and  that  the  several  quotients  are  the  partial  denomi- 
nators of  the  continued  fraction.  He  will  find  this  statement  to 
contain  a  convenient  working  rule  for  converting  common  frac- 
tions into  continued  fractions. 


1.  §2.]  CONVERSION  OF  CO^EVION  FRACTIONS.  139 

So  the  value  of  tt,  the  ratio  of  the  ch-cumference  of  a  circle  to 
its  diameter,  is  3.14159  26535  •••.  If  in  place  of  this  endless 
decimal  3.1416  he  used,  then 

-  =  3Am  =  3  +  i-^  =  3+l-    1      =3  +  i-i 

and  its  convergents  are  :  11 

3;    3  +  i,  =  3^,=^;   S+l-_j_,=S^,  =  ^;  .... 

16 
Note.     The  real  value  of  tt  is  incommensurable ;  but  if  the 
decimal  be  taken  to  20  places,  then  the  partial  denominators  are 
7,  15,  1,  292,  1,  1,  1,  2,  1,  3,  1,  14,  2,  1,  1,  2,  2,  2,  ...,  the 
continued  fraction  is 

3+i-    1 

292  +  ..., 
and  Its  convergents  are  : 

3,  3f,  3^^,  3JA,  3^^,  .... 
(6)  Numerical  or  literal,  ni,  n2,  •••  any  entire  numbers: 
Reduce  to  a  proper  fraction  or  mixed  number;  divide  as  above 
(a) ,  except  that  factors  may  be  stricJien  out  of  the  numerators 
(divisors)  or  introduced  into  the  denominators  (dividends)  and 
reserved  as  partial  numerators  of  the  continued  fraction. 

T^      ,,     I  striking  out     «      «    .       i  from  a  divisor    . 

^°'  ^''-i  introduction  °^  ^  ^^"^'^  Into  a  dividend  ^^  ^l""'^" 
lent  to  dividing  both  terms  of  the  fractional  part  by  the  ratio  of 
its  numerator  to  this  factor. 

^  '     101       710.       7  —      7-2  1 

J.U1  <  j3  /   t^3         '9-1 

wherein  the  factor  2,  stricken  out  from  the  first  two  divisors, 
26  and  10,  and  introduced  into  the  third  dividend 
5,  becomes  the  numerator  of  the  first  three  partial 
fractions ; 

or,  both  tei-ms  of  y^^'V  ^^^  divided  by  13,  those  of  f|  by  5, 

and  those  of  f  by  f . 


140  CONTINUED  FRACTIONS.  [VI.  pr. 

o  630       2  2  2 

wherein  the  divisors  are  315,  47,  7,  the  reserved  factors  are  all  2, 
and  the  quotients  are  so  taken  that  the  remainders  are 
negative. 

24+6a;  2H 2H x 

^12+a;  3  +  p 

4 

wherein,  at  each  division,  the  factor  x  has  been  stricken  out  of 
the  divisor  and  reserved  as  the  numerator  of  a  partial 
fraction ; 

and  the  convergents  are :         ^ 

X  _2x        X  _  6a;  +  a^ 

'''    r  +  ?'"2T^'     FT— a;'"6T4^'  "*' 

2  ^+i 

In  this  example,  if  a?  be  small,  the  successive  convergents  rap- 
idly approach  the  true  value  of  the  fraction. 

aa:^+  ac  I  x 

-\-ad\  <^T.9T.j<^;  ^ 


So 


x'-\-b 
-he 


bx^-{-bd=—,b  =— ,  & 


x'-^-bd     x-\--j- — p-      x-\ —  ex         ^-\ :c      , 


X 

wherein,  at  the  successive  divisions,  the  factors  a,  b,  c,  d  have 
been  stricken  out  of  the  divisors  and  reserved  as  the 
numerators  of  the  partial  fractions ; 
and  its  convergents  are  : 

a  ax  ay?-\-  dc 


X         ar^+6'        a;^H-6|a;' 


In  this  example,  if  a;  be  large,  the  successive  convergents  rap- 
idly approach  the  true  value  of  the  fraction. 

Note.  The  continued  fractions  presented  in  this  problem  are 
all  finite,  and  the  original  fraction  is  the  last  convergent ;  those 
which  follow  are  infinite. 


2.  §3.]  CONVEESION  OF  SURDS.  -  141 

§3.     CONVERSION   OF   SURDS. 

PrOB.  2.     To   CONVERT  A  SURD   OF  THE  FORM  •^(a?±h)  INTO  A 
CONTINUED   FRACTION. 

Let  X  =  the  value  of  ■^{o?±h)  ; 

then   •.*  a^  =  a^±6,  whence  a^—a^=  ±6  and  x  —  a=   ^  ^  , 

a-\-x 

1]       .-.  a;  =a±— ^  =  a±5 5        =a±-A—  5 

■■  a  +  x  a  +  a±- 2a±- 

T  a-{-x  a  +  x 

2a±-^— 

2a±..., 

and  its  convergents,  if  the  entire  number  a  be  included,  are  : 

a;   a±A,=2a^;   « ± -^    5,  =  '"^^'^^  .... 
2a  2a  2a±~'        4a2±6   ' 

2a 

^.^.,    V2=V(1+1)  =  H-^1 

24-  — 
and  the  convergents  are :  2  H 

1'  H^  If?  1-A-'  •••?   =1'  2'  5'  12'     '"' 
whose  squares  are : 

1,  1,  49,  289  .  _       +1     _1         J_   _ 
4     25    144'  4         25  144' 

So        y'3  =  V(4-l)  =  2-i— 1 

*-4^1_ 
and  the  convergents  are  :  4  __  ...^ 

9     7     26     97 

'   4'    15'   56'  *"' 
whose  squares  are : 

^     49     676     9409 

'    16'    225'    3136'  '"' 

=  ^  +  ^'^  +  ^'^+2l5.'^+3^'"-- 
So        V7  =  V(4  +  3)  =  2  +  f-.  3 

and  the  convergents  are :  4  +  •  •  • , 

2,  2f,  2«,  2«,...,  =2,   11,   |,  ^,..., 


142  CONTINUED  FRACTIONS.  [VI.  pr. 

whose  squares  are : 

.  121  2500  54289  707,9-      27    „  ,    81 


16'  361'  7744'      '  '        16'        361'        7744' 

Or     V7  =  V(9-2)=3-?-2     ^     =3-i-i 

and  the  convergents  are :  6—  ••♦,  3  —  ••• 

o     16     90     508  _q     8     45     127 

'    T   34'    192''"'  "^'   3'    17'   Is"'"*' 

whose  squares  are : 


9'   289'   2304'     '         '    '      '9         289         2304'      ' 

Note  1.  The  rule  is  given  in  formula ;  the  reader  may  trans- 
late it  into  words.  In  general,  he  will  find  any  such  formula 
translatable  both  as  a  theorem  and  as  a  rule.  The  first  is  a 
statement  of  facts  and  is  put  in  the  indicative  mood ;  the  other  is 
a  direction,  an  order,  and  is  put  in  the  imperative  mood. 

Note  2.     If  —  be  small,  the  errors  of  the  squares  of  the  suc- 
ar 

cessive  convergents,  and  therefore  of  the  convergents  themselves, 
diminish  rapidl}'. 

For     (a)^  =  a^-(a2±6)  =  6. 

^      [-2^}^ i^ =  "^'  +  4^^' 

^0?  Aa^ 


So 


/4a»±3a5Y 


'4a8±3a5Y  _  16a«±  24a^6  +  9a'6« 
16a^±8a26  +  62 


=  a'±b^: ^ 


16a^±8a26  + 


('''^^)=16a^±L.+5^=^-(^^J'C^-^-)- 


2.  §3.]  CONYEESION  OF  SURDS.  143 

So        in  the  above  numerical  examples,  the  smaller  —  is.  the 

(J?     ' 
more  rapidly  does  the  series  converge. 

E.g.,    the  series  got  by  taking  V7  =  V(^- 2),  wherein -=  ?, 

a-  9 
converges  much  more  rapidly  than  that  got  bj^  tak- 
ing V7=  V  (4 +3),  wherein  ^  =  -. 

or     4 

So        V3  =  VC"^""!)  gives  —=-5  and  the  square  of  the 
fourth  convergent  differs  from  3  by  only  rr^  ; 

whereas   V^  =  V(  1  + 1 )  gives  -  =  -,  and  the  square  of  the  fourth 

(XX  ■< 

convergent  differs  from  2  by 

144 

Note  3.     Another  conversion  may  be  made  thus  : 

•••  x^—a^=±b,       whence  x-\-a  = 
on  b  ^ 


a  —  x 


a-{-a± 


a—x 
b  b 


a  —  x  2a±f, 

wherein   the  convergents  are  the  opposites,  each  of  each,  of  those 

found  b}^  the  first  conversion  ; 
i.e.,  bj'  the  first  process  the  convergents  of  the  positive  root 

were  found,  and  by  the  second,  those  of  the  nega- 
tive root,  equally  large  but  of  contrary  sense. 

E.g.,    V3  =  V(4-l)  =  -2+i-i_ 

and  the  convergents  are  4--"«, 

;.         7         26         97 
~''   "T    ~15'    "56'"" 
^P*  For  other  uses  of  continued  fractions  see  the  computation 
of  logarithms  [IX.  §  3] ,  and  the  solution  of  quadratic  equations 
[XI.  §  13]. 


144  CONTINUED  FRACTIONS.  [VI.  pr. 


§  4.     COMPUTATION   OF   CONVERGENTS. 

PrOB.  3.    To  FIND  THE  CONVERGENTS  OF  A  CONTINUED  FRACTION. 

First  Method.  For  the  first  convergent  reject  all  after  the 
first  partial  fraction^  for  the  second  convergent  reject  all  after  the 
second  partial  fraction^  and  so  on;  reduce  to  simple  fractions 
the  complex  fractions  that  remain. 

The  examples  given  above  have  all  been  solved  by  this  method. 

Second  Method.    Form  two  series,  -l  ^^'  ^-'  ^^'  ***  ^'" 

'  Di,   Dg,   Dg,  ...  Dk, 

Wherein  i"^^^^-       ''^^S^^^       =Ni-d., 

'Di  =  di,       D2  =  did2  +  n2  =  Di.d2  +  n2, 

.  N3  =  nid2d3-}-nin3  =N2.d3  4-Ni.n3,  ••. 

'  D3  =  did2d3  +  n2cl3  +  din3  =  D2-d3  +  Di.n3,  .•• 

31  J  Nk  =  Nk_i  •  dk  +  Ni,_2  •  Hk  . 

J  ^  Dk  =  Dk_i  •  dk4-Dk_2  •  nk ' 

then  are     ^,  ^,  ^,  ••.  ^,  =  ^^-i'^^  +  ^t-2'n^^ 

Di    D2    Dg  Dk  r>k-l  •  tlfc  +  I>k-2  •  Hk 

the  Ist^  2d,  3d,  .••  k^^,  convergents. 

The  reader  may  translate  this  formula  into  words. 

1 .    The  law  is  true  for  the  third  convergent. 
For      the  first  three  convergents,  hy  the  first  method,  are 
^1         ^'^^-         and  n^d^-d^  +  nyn^ 


di     d^d^-^-  n2  di  c?2 •  c?3  +  n2 -  d^-^-  di'  n^ 

^N2.d3  +  yi-n3^  as  above.  q.e.d. 

D2  •  C?3  +  Di  •  7l3 

2,   If  the  laio  he  true  for  the  'kth  convergent,  it  is  true  also 
for  the  (k  +  l)i/i  convergent, 

p_      .  .  Nj    _  Til  _  N,.i.(^;,+  N,_2.n;,  p, 

F^^    •  D,'  =  ^  ^       ^;  =  D,.,.d,+  D,.2.n;       Lhyp. 

^2  +  ...+- 

an  identity,  whatever  expression  or  value  dj,  may  stand 
for,  and  therefore  an  identity  when  dj,  is  replaced 

byc?,+^. 


3.  §  4.] 


COMPUTATION  OF  CONVERGENTS, 


145 


^2+-  + 


d,+ 


^k+i 


du  + 


\  (^k  4  1/ 

(»*-!  •  f^*+  I>*-2  -7**)  •  (^i+1  +  I>*-1  •  W;fc  +  1 

Q.  E.D. 

3.   !Z%e  Zaz/;  is  true  whatever  the  index  k. 
For     •••  it  is  true  for  A;  =  3, 
.*.  it  is  true  for  k  =  4. 
So  for  /c  =  5  ;  for  ^'  =  6  ;  for  A;  =  7  ;  and  so  on.   q.  e.  d 

E.g.,    of  the  fraction  —  t, 

x+—  c 

the  convergents  are  :  x-\ , 


^k-di+i-\-Bu-i"nu+i 


[1 
[2 


ax 


aX'X-{-a'C 


aa^-i-ac 


+  c 

ax^  +  ac 
(ax^  +  ac)  '  X -\- ax '  d  -j- ad 


fa^  +  b\x\ 


^a^  +  b\x\       ,  /  o  ,  ,,    ,       a;*4-& 


+  c« 


x'-i-bd 


So        of  the  fraction  —  x 

the  convergents  are  :  c+  •••, 

»      bx  x^  -\-bcx  (b  -\-d)x^-\-bcdx         .  ^ 

a  x-\-ab  {a+c)x-\-abG  x^-\-{ab+ad-\-cd)x+abcd 


146  CONTINUED  FEACTIONS.  [VI.  pr. 

In  particular,  when  ni,  ng,  •••  n^,  eacJi  =  1,  then  the  formula 
becomes : 

The  reader  ma}'  translate  this  formula  into  words.  He  may 
also  demonstrate  it  anew,  putting  1  for  Wj,  W2,  •••  n^,  nj^+i,  in  the 
general  demonstration. 

JE.g.,   of  the  fraction  —  1 

the  convergents  are :  c  -f  •  •  •  > 

1  b  6.c-fl  bc+1 


a*  aft  +  l'    (a6-fl)-cH-a       a&c  +  a  +  c' 

Compare  the  result  of  the  previous  example,  when  x  =  l. 

Note.     Formulae  [3,  4]  may  be  made  to  include  convergents 

of  the  mixed  number  Uq  +  ^^  ^o 
di-\ — ^— 

(^2+  •••?  ^3  follows  : 

Let      ^^^Zn'  !!"-?' 

»  D_i=0,    Do=l, 

wherein  Uq  =  any  number,  perhaps  0 ; 
then  wiU^  "''  =  "»•'!' +  "-'■""    ''2  =  N,-d2  +  N„-», 


and 


Di  =  Do  •  cZi  +  D_i  •  ?li,      D2  =  Di  •  cZg  4-  Do  •  7l2, 
Ni  _  No  •  di  +  N_i  •  Til  N2  _  Ni  •  C?2  +  No  •  W2 

Di^Do-di+D.i-rii'  D2"~Di-cZ24-Do-W2 

as  the  reader  may  verify  ; 


but  it  is  convenient  to  speak  of  — ,  =  Wo+  37  =  -^-^ ->  ^ 

Di  «!  "1 

the  1st  convergent,  even  when  Wq  has  a  value  not  0  ; 


and 

of  —  and  —  as  the  0th  and  ~lst  convergents. 

E.g., 

1                                                             N          3 

of  TT,  =3-  1     ,  the  convergents  after  — ,  =t?  are 
1... 

3.7+1   _22.  22-15+3  _333.   333-1+22  _  3.55 . 

l-7+0'~7'     7-15  +  1' ~106'    106-1+7'      113' 

... ;  as  in  §  2. 

1.  §  5.]  GENEEAL  PEOPEBTIES.  147 

§  5.     GENERAL   PEOPERTIES. 
In  this  section,  let  Vi,  Vg,  Vg,  •••  be  the  true  values  to  which 

_  _    N;i  _    Til 

are  convergents ;  whence 

Til  Tlj  

Theor.  1 .  i/"  <^6  partial  numerators  and  denominators  be  all 
positive,  the  convergents  Ci^i,  Ci,2>  •••  are  alternately  greater  and 
less  than  the  true  value  Vi. 

Let      Tij,  W2,  •••  c?i,  c?2,  •••  be  all  positive  ;  then  will 

Ci>Yi,    C2<Vi,    C8>Vi,...C,^>Yi,^^^^^;^ 

For      !Li>^^,  [11.  ax.  18 

di       c^i  +  V2 

i.e.,  Ci  >  Vj.  Q.E.D. 

I.e.,  C2  <  Vi-  Q.E.D. 

C^3  ^3  +  ^4  "2+3-        (^2-+ 


d^  '        d3  +  Y4 

•    •    -^,    W2  >^    ^2 

oi  + ;,— L  **3    ^1  +  ;rx  !^3 

C?3  C«3  4-  V4 

I.e.,  C3  >  Vi.  Q.E.D. 


[     UN 


or  THE  A 

'NIVERSITY  1 


148  CONTINUED  FRACTIONS.  [VI.  th. 

Theor.  2.    If  any  convergent^  c^,  he  subtracted  from  the  next 
following  convergent^  Ck+i,  then  will  the  remainder^ 
^-1  Nk+i      Nk  ni-no'-Hk  +  i 


I>k+1     I>k     ^      '       I>k-I>k+] 

N 


Since — -  = — •>    it  IS  to  prove  that 

I>**N*  +  i  — N4.D4  +  I  =  (  —  !)*.  711-712  •••n^fc^.i. 

1.  The  law  is  true  for  k  =  1 . 

For     • .  •  Ni  =  Til,     Di  =  C^i,     N2  =  Ui  c?2,     D2  =  cZi  c?2  +  W2, 
.•.  Di'N2—  Ni-D2=  c?i«nidJ2  — Tli-('^1<^2  +  ^2) 

=  — 7li«?l2  =  (  — l)^*'^l'^2-         Q.E.D. 

2.  7/"  ©k-l  •  Nk  —  Nj,_i  •  Dk  =  (—  1)^"^  •  Di  •  112  •••  Ilk,     ih^l^   f^itt 

l>k-Nk+i-Nk-Dk+i  =  (  — l)^-ni-n2---nk+i. 

For      J)k'^k+i-^k''Dk+i='Dk'('ifk'd,+i-h^k-i-nk+i) 

-N,.(D;fc.d;fc  +  l  +  D,_l.Wjfc  +  l)  [3 

=  (N*_i  •  D;fc  —  D^fc.l .  Njfc)  .  7I4  +  1 
=  -  (d,_i  .  N*  -  N,_i  •  Di)  •  W,  +  i 

=  -(-l)*-i.?ii.W2  —  n;,.n,+i    [h^-p. 
==(  — l)*-ni-n2---n44.i.     q.e.d. 

3.  The  law  is  true  whatever  the  index  k. 

For     *.•  it  is  true  for  A;  =  1,  [1,  above 

.*.  it  is  frue  for  A;  =  2.  [2,  above 

So            forA;  =  3,    forA:  =  4,   forfc  =  5,  •••.  q.e.d. 

In  particular:   If  Wi,  ng,  •••  each  =  1,  then  will 

(-1)* 
7]  c,^i--        ^       ^ 


Dt'D 


*  +  l 


Cor.  1.     TJie  error  of  any  convergent,  — ,  of  j~4_l 

when  di,  d2,  •••  are  all  positive,  *^2+  *••» 

IS  less  than ,  and  much  more  is  it  less  than  — -. 

i>k-i>k+i  i>k 

For     •••  the  true  value  lies  between  —  and ,  [th.  1 

i>*  i>*+i  "■ 

and     •.*  these  differ  by  only ,   <— 5, 

i>*'i>*+i        ^k 

.'.    etc.  Q.E.D, 


2.  §  5.]  GENERAL  PROPERTIES.  149 


111     Hi  •  no     111 .  1^2  •  ns 
Cor.  2.   c,=  ^^     ^^.^+    „^.^^       • 

"  +  {-ir'' 

111  •  n2  •  •  •  Dk 
i>k-i-i>k 

?ll 

)r     •••  Ci            =— , 
i>i 

Til  •  rig  r  , 

Cg  —  Ci    = i — -i  ph. 

^1  •  '^2  •  ^3 


Co  —  Co      — 

D2-D3 

and  c,-C;t_i  =  (-l) 


-J  *   •      *        Di        Di-D2  D2-D3  '"       ^  ^        '     Dt_i.DA 

Jn  particular:   If  nj,  rig,  •••  each  =  1,  then  will 

-■  *        Di        Di-Dg        D2-D3  ^A-l  *  Dj 

Note.   This  formula  gives  a  rule  by  which  any  continued  frac- 
tion may  be  reduced  to  a  series. 

E.g.,    V28=V(25  +  3)  =  5+^3_ 

10+..., 
whose  1st,  2d,  ...  convergents  have  the  denominators 
10,   103,   1060,   10909,   112270,  ... ; 

q  02  03  Q4 

.-.  V28  =  5H-i +        "^  "^ 


10     10.103     103.1060     1060-10909         ' 
a  series  whose  successive  terms  grow  smaller  very  rapidly. 

CoR.  3.    If  ni,  n2, ...  dj,  da,  .••  be  all  positive,  the  successive 

differences  grow  smaller  and  smaller. 

,-,                                 n^n^-'-nj,                          Wi7?2--- w*w*+i    rp^  1 
tor    •.•  c,_i'-c,  =  — — andc,'-c,+i= — »  Lcr-^ 


and 


ninz-'-n^                          Wi7?2--- w*w*+i 
: and  Cv'^Cj.+i— 

rii-nz ...  n,,  ^  Tii.ng...  nj^-nj^+i _      Da+i 
i>*-iDa      •  i>*.i>A+i         ""  i>*-i  • '^*+i 

~         I>*-l-^*+l 

Ct-i-C*  ^  C*'-C;,+i.  Q.E.D. 


[3 


150  CONTINUED  FEACTIONS.  [VI.  th. 

Cor.  4.  The  successive  convergents  approach  to  each  other, 
and  therefore  to  the  true  value  which  lies  between  them,  rapidly 
when  the  ratios  di •  d2 :  ng,   dg •  dg :  Dg,  •••   are  large. 

For      ^^^^* ,  =  1  +  J^t:£^±L,  [above 


=  1  + 


JF-gr.,    the  convergents  to  V(a^  +  6),   =a-f 


2a  + 


7,  2a  +  ..., 

approach  the  trae  value  rapidly  when  —  is  small.  [§  2 

a 

CoR.  5.    jy  ni,  nj,  •••  eac^  =  1,  and  dj,  dg,  •••  6e  all  entire,  all 

the  convergents  are  simple  fractions  in  their  lowest  terms,  and 

T  numerators 

their  consecutive  \  ^^..^^i^^^^^^  are  prime  to  each  other. 

For     •••  every  common  measure  of  n^,  d^  is  a  measure  of 

^k''Dt  +  i-^-D„-N^+i,  =  1.  [III.  th.2  cr.4 

.*.  the  h.  c.  msr.  of  n^,  d^  is  1.  q.  e.  d. 

So  of  Ni,  Nj+i,    and   of  D;^,  d^+i. 

Cor.  6.  Ifui,  Ug,  •••  each  =  1,  and  di,  dg,  •••  be  positive,  then 
between  two  consecutive  convergents  there  lies  no  fraction  whose 
denominator  is  smaller  than  the  largest  of  their  denominators. 

For,  let  -  ,  4^—,  ~  any  simple  fraction  wherein  d  <  d.  . . ; 


then 


N        Nfc        N  •  Dj  '^  D  •  Nju 

—  r^  =  , 

whose  numerator,  being  entire  and  not  0,  [11.  ax.  23,  hyp. 

either  :=:  or  >  1, 
and  whose  denominator  <  d^  •  d^+j,  [hjT). 

...  ^_^>_i ;  [n.  ax.  17, 18 

i.e.,         -  differs  more  from  ^  than  ^^i±l  does, 

and  cannot  lie  between  them.  q.  e.  d. 


3, 4.  §  5.]  GENERAL  PROPERTIES.  151 

Theor.  3.    The  difference  ^^-^^  (-l)^n,...  n,^,.d,^,^ 

I>k  +  2         I>k  r)k -01^  +  2 

I>A  +  2  I>*~"  ^k'^k  +  2 

whose  numerator  =     ^k' (^k+i' <^k+2  +  ^k' n^+2) 

=  {-iy>ni"'ni,  +  i'd,   2,  [th.2 

•  ••    etc.  Q.E.D. 

Cor.  1.    i/"  ni,  Dg,  •••  and  di,  dg,  •••  6e  all  positive, 
then         Ci  >  Cg  >  Cfi  >  •-., 
aijc?  C2  <  C4  <  Ce  <  •••. 

Cor.  2.    y,  =  n,  +  ^  +  "-i^i^  +  ^i:::^  +  ... 

I.D2  D2-D4  D4-D6 

Ni         7li7l2^3        Wi«-«  7146^5 


Di  Di  •  Dg  Dg  •  D5 

Theor.  4.    Jn  an2/  continued  fraction 

I)k-(l>k  +  l+I>k-Vk  +  2) 


For 


Til  _  nj 

.'.  Vi  is  what  C;fc+i becomes  when  for  c^^^+i  is  put  f^ft+i+v*+2» 
and    •■•c>,.-c,=  (-^);-"--"'-' 

i>*  •  (i>*- ^A + 1  +  i>t-i  •  ^A + 1) ' 

(-l)*.ni...?2;,  +  l 


152  CONTINUED  FRACTIONS.  [VI.  pr.  4. 

Cor.  1.     If  Ui^  ng,  •••  each  =  1,  and  d^+s^  d^+g,  •••  each  <  1, 
then         Vi  '^  Cfe  ?ies  between  and 


i>k-Dk+i  i>k-(Dk+i+i>k) 

For     •.•  Vi4.2,  =  J''^^  »  then  lies  between  0  and  1, 

.••  D4«(D;fc^.i4- D;k«Vj+2)>  the  denominator  of  Vi~C;fc,  lies 
between  d*  •  d^ + 1  and  d*  •  (D;^ + 1  +  D;^)  ; 

.-.    etc.  Q.E.D. 

Note.    So,  if  of  ni,  Wg,  •••  any  are  negative  but  all  ih  1,  and 
if  c?i,  (Z2,  •••  <  2,  then  Y^-i-i  lies  between  —  1  and  -{-  1  inclusive, 

and  Vi '~  Cfc  between and 


CoR.  2.    7/*  ni,  ng,  •••  eac7i  =  1,  a7id  di,  dj,  •••  eac/i  <  1,  every 
convergent  differs  less  than  the  previous  one  from  the  true  value. 


For 

•••   D*+2  =  I>*+l-C^*+2  +  »*  <!>*+! +  !>*> 

[^*  +  2<l 

and 

•••  r>*+i-»*+2>i>*-(i>ik+i4-i>»)  ; 

and 

.(D 

^          „   [cr.l 

•••    Cjk  +  i~Vi<Ci~Vi.  Q.E.D. 

Prob.  4.    Given  an  incommensurable,  ic,  and  an  integer,  i : 

To    FIND    A    simple    FRACTION    -     WHOSE    DENOMINATOR    IS    SUCH 

N  ^     1       " 

THAT  D  3^  i   AND    -  ~  a;  <  • 

D  D'i 

Reduce  xto  a  continued  fraction  wherein  ni,  Ug,  •  •  •  are  each  1 , 

and  di,  dg,  •••  are  all  positive  integers;  find  — ,  the  highest  con- 

i>k 
vergent  whose  denominator  ^  i;  it  is  tJie  fraction  required. 

For     '.'  x^-'< — ,  [th.2cr.l 


D^        D^ . D 


k  +  1 


and     •.*  Dj+i    >i'. 


.  a;'^-< ;»       i.e.,  «^ — ;•  q.e.d. 

Di      Di  •  I  D  •  t 


th.5.  §6.]  SECONDARY   CONVEEGENTS.  153 

§  6.     SECONDARY    CONVERGENTS. 

If  Ml,  no,  •••  each  =  1,  and  dj,  do,  •••  be  all  positive  integers, 

then  the  series  Cq,  Cj,  C2,  Cg,  •••  may  be  resolved  into  two  series : 

Gqi  C2,  C4,  •  •  • ,  all  too  sriiall,  and  Cj,  Cg,  C5,  •  •  • ,  all  too  large, 

d  d 

wherein   Cg  — Co  =  — —^      04  —  09  =  — —->  "•■>  [th.  3 

Do  •  D2  D2  •  D4 

,                           —ds                        —ds 
and  03-Ci  = i,      €5-03  = ^,  •••. 

Di-Dg  D3.D5 

Put       1,  2,  3, ...  cZa-l ,  in  turn,  for  cZg  in  ^I'^^  +  ^o,  =  Cg,     ' 

Di .  (^2  4-  i>o 
and  1,  2,  3,  •••  c?4— 1,  in  turn,  for  d^  in  O4,    and  so  on  ; 

then         a  series  of  secondary  convergents  is  formed,  Ijing  be- 
tween the  primary  convergents  Oq,  C2,  C4, ...  ; 
and  the  whole  series,  whose' terms  are  all  too  small,  is  : 

^   Ni  +  Nq  2.^1+  Nq  3.Ni  +  yo     (d2-l)-Ni+No  ^^ 

*"  Di  +  Do'  2.D1+D0'  3.D1  +  D0'*"  (C?2— l)-Di+Do'  ■' 

N3  +  N2  2  .  N3  +  N2  3  .  N3  +  N2     ((^4-l).N3+J^2  ^  . 

D3  +  D2'  2.D3+D2'  3.D3+D2'***  (d4-l).D3+D2'  '^ 

or        Co?  Co|i,  Co  I  2?  •*•  ^0\d^-li   C2,  C211,  C212?  •••  G2\d^-l^>   C4,  ••'. 

So, put  1,  2,  3,  •••  c?3— 1,  in  turn,  for  d^  in  O3, 
and  1,  2,  3,  ...  d^—l,  in  turn,  for  d^  in  O5,    and  so  on  ; 

then         a  second  series  is  formed,  whose  terms  are  all  too  large  : 

Ns  +  Ni       2.N24-N1  N4+N3       2.N4+y3 

D2  +  D1       2.D2  +  D1  U4+D3       2.D4+D3 


■-5? 


or       Ci,  Ci|i,  Ci|2,  •••  Ci|^^_i,  C3,  C311,  C312,  •••  Cg\a^_i,   C5 

Theor.  5.     The  terms  of  the  first  series  Oq,  Co|i,  Co|2?  •••  grow 

greater  and  greater;  of  the  second  series  Oj,  Ci|i,  Ci|2,  •••  less  and 

less  ;  and  the  differences  of  successive  terms,  smaller  and  smaller. 

For      Oo„-0o    ^;^o_No^  1  ^^j^2 

°"  Dl  +  Do        Do        Do.(Di  +  Do) 

Co|2  — Co|i  =  ;; ; r— ; r,     and  SO  On. 

(Di  4- Do) '(201  + Do) 

C^  ^  r.  N2  +  N1        Ni  -1 

bo  Ci  1 1  —  Ci      = ; =  7 ; r  ' 

"'  D2  +  D1        Di         Di-(D2  +  I>i) 

Ci|2-Ci|i  =  ; ; ^-- ; r,    and  SO  OH. 

(D2  +  Di).(2D2  4-Di) 


154  CONTINUED  FRACTIONS.  [VI.  tli. 

Theor.  6.  Ck  I  r  '^  Ck+ 1  = ,  wherein  Dk  | ,.  =  r  •  Dk+i+ Dk- 

For,  reduce  c^i^'^  C;t^.i  to  the  common  denominator  -Dj^.^-Tf^; 
then         the  numerator,  (r«Nj+i+N;fc)«D4+i~(?'-D;t+i+i>it)*N4^i, 

=  ^k''^k  +  l^^k'^k+l^   =1-  Q.E.D. 

Cor.  1 .     Ck  I  r  «s  in  its  loivest  terms. 
For     •.•  ever}'  common  measure  of  N;^!^,  d^,^  is  a  measure 

of  Niir-i>t+i~i>*ir-N*  +  i.  =1.    [III.  th.2cr.4,th. 
.*.  the  h.  c. msr.  of  Njn ^,  Dt I ^  is  1 .  q.e.d. 

Cor.  2.     Between  Ck|r  (md  Vi  there  lies  no  simple  fraction  with 
terms  so  small  as  those  o/  Ck  |  ,- 

For     •••  c,,,'-c,+i  =  - -—., 

.*.  between  c^i^  and  Cj^+i  there  lies  no  simple  fraction  with 
denominator  so  small  as  D;^  |  ^ ;  [compare  th.  2  cr.  5 
and     *.*  Vi  lies  between  C;t I ^  and  Ci+i, 

.*.  between  c^i^  and  Vj  there  lies  no  simple  fraction  with 
terms  so  small  as  those  of  c^ | ^.  q.e.d. 

Theor.  7.     Ck|r'^Ck|B  = 


^k  I r  '  ^k  I  8 

For,  reduce  c^t ,  ^  ~  c^  | ,  to  the  common  denominator  d^  ,  ^  •  d^^  , , ; 
then  the  numerator, 

(Njfc+i  •  r  +  N;,) .  (d,.^.i  .  S  +  D,)  ~  (D;t+i .  r  +  D,) .  (n,+i  .  .S  +  N,), 
=  K.D,+i'-D,.x,+i)-(r~s)=r~s.  Q.E.D.  [th. 2 


Theor.  8.    When  r  >  •|-dk+2?  t^^i^  Ck|r  differs  less  from  Vi  than 
does  any  simple  fraction  with  terms  as  small  as  those  of  Ck  |  r- 

Di  +  K  +  1  4-  •••)•»*  +  !' 


For  '•'y.=::iy7^'T::i[y\^c,,.,     [^<cz.^.+i 


and     •.•  Ci,^~C;fc|,  = <  7 ; r [r>^s 

'"^  *"        D,,,.D„.         (r.D,  +  i  +  D,).D„.       L     ^12- 


1 


Cjfci^  is  nearer  than  0^4.1  to  Vi.  q.  e.  d. 


6-8.  §7.]  EXAMPLES.  155 

§  7.     EXAMPLES. 
§§  2,  4.      PROBS.  1,  3. 

1.  Convert  the  following  fractions  into  continued  fractions, 

and  get  five  convergents  for  each  of  them,  if  there  be  so 
many: 

47_   293   839   995   1051   2371 
223'   631'   739'   293'   237l'   4049* 

2.  Find  the  limits  of  error  of  the  fourth  convergent  of 

.1357,    2.7182818,    .43429448,    180:57.2957795. 

3.  The  true  length  of  the  equinoctial  year  is  365^  5^  48'"  46' : 

reduce  the  ratio,  5*^  48™  46'  :  24'\  to  a  continued  frac- 
tion, find  five  convergents,  and  thence  show  how  often 
leap-3'ear  should  come.  Find  the  limits  of  error  for  the 
fifth  convergent. 

4.  The  earth  makes  one  sidereal  revolution  about  the  sun  in 

365.2564  days,  and  Venus  in  224.7008  daj's  ;  how  often 
will  the  two  planets  be  in  conjunction  (in  line  with  the 
sun)  ?  Get  four  convergents,  and  the  limits  of  error  for 
the  fourth. 

§§  3,  4.    PROBS.  2,  3. 

5.  Convert  the  following  surds  into  continued  fractions,  and 

get  five  convergents.     Find  the  limits  of  error. 

V(a'-l),    V(«'+«)'    V(«'-^)'    V(«'+a+l). 


6.   Express  y'c,   =  V(^^  +  c  —  m^) ,  as  a  continued  fraction, 
and  show  that  whatever  the  numbers  m,  m', 

m+- -c-m''     =±m'±  c-m" 

2mH 2m'+ 


2m  +  ."  2m'+---- 

7.   Develop  into  series  the  sixth, convergents  of  [th.2  cr.2 

3.U159265,  |1|,   V145,  ^| ;    V(«'+«').  V(«=-»)- 


156  CONTINUED  FRACTIONS.  [VI. 

8.    Show  that: 

1    rii  Ui 

a2  H tta :  7?i  +  -5 :  Jh 


do 'in 


c?^+-  '     -(^2+ 


^3  + 


d"-%--  ^d"+     '' 


d'" -d"'-{-"-. 

355 

9.   Write  all  the  primarj'  and  secondary  convergents  of in 

113 

order ;  and  find  the  fraction  that  differs  least  from  it  of 

all  those  whose  denominators  ^100.  [§  6,  th.  6  cr.  2 

10.   Find  all  the  fractions  -  such  that  d  ^  50,  -  ^  —  <     ^ 


D  D      113      50. D 

1 1 .   Prove  that     ^  ^*  =  {n,  +  d,.,'  d,)  {  I'-'  +  n,.,  •  d,  {  l^'^' 


12.    The  continued  fraction  - 1  ^  occurs  in  botanj^,  zoology,  and 


1 

1... 
How  does  this  fraction  differ  from 


astronom}'  1 


4 


Find  twelve  convergents,  and  prove  that :  ^ 

Ni=        2    .Nt_2       +1        .N*_3=... 

=        N^  •  Ni_^^.  2—  N^_2  •  ^k-r  -^ 

=  i(Nr  •  N;,_^  +  3+  N,-3  •  ^k-r)  ,      Whcrciu  ^  =  — 

that  .*.  N^  measures  Ng^,  Ng^,  •••  ; 

and  that  N^  •  n,+,  —  nv+,  •  n,  =  (  —  1)'n^_,  •  n,    if  r  >  t. 

13 .    Convert  the  series  ao + «o  «i  ^ + «o  <^i  <^2  a^^  H —  ±  ao  •  •  •  a„  a?"  into 
the  continued  fraction,  and  find  its  first  four  convergents, 

1 TT—    ^2^ 

ChiC+1 = -—  „   „ 

^    ^         a,.x-\-\-.._     «na?  ,  r§2 

a„a;4-l  "^ 


§1.]         VARIABLES  AND  CONSTANTS.  -  CONTINUITY.         157 


VII.     INCOMMENSURABLES,  LIMITS,  INFINITESI- 
MALS, AND  DERIVATIVES. 

§1.     VARIABLES    AND    CONSTANTS.— CONTINUITY. 

When  the  conditions  of  an  investigation  are  such  that  one 
number  takes  a  series  of  different  values,  that  number  is  a  vari- 
able;  but  a  number  that  keeps  its  one  value  unchanged,  is  a 
constant;  and  the  particular  values  that  ma}^  be  given  to  vari- 
ables are  constants.  The  remainder  got  by  subtracting  one  value 
from  the  next  is  the  increment  of  the  value  first  named. 

When  one  variable  is  a  function  of  another,  the  first  is  a 
dependent  variable,  and  the  other  is  an  independent  variable. 
From  the  fixed  values  of  the  constants  and  the  values  that  may 
be  given  to  the  independent  variables,  the  corresponding  values 
of  the  dependent  variables,  or  functions,  may  be  computed. 

E.g.^  while  a  sum  of  money  remains  at  interest,  the  principal 
and  rate  are  constants,  but  the  time  and  accrued  interest  are 
variables,  of  which  either  may  be  taken  as  the  independent 
variable,  and  the  other  is  dependent  upon  it ;  for  when  the  prin- 
cipal, rate,  and-J  !^"f^     ,   are  given,  the^  interest  .^  ^i^^^^^^    ^jg. 

^     '         '  '  ui  to  rest  ="        '         '  time 

termined ;    and  to  different  values  given  to  -{  ;.-.fgj.ggf  different 

1  r,  f  interest  , 

values  oi  ^  , .  correspond. 

So,  the  radius,  circumference,  and  area  of  a  circle  are  all  func- 
tions of  each  other,  and  all  grow  together  if  the  circle  increases  ; 
but  the  ratio  of  the  circumference  to  the  radius  is  constant,  and 
so  is  the  ratio  of  the  area  to  the  square  of  either  of  them. 

When  the  variable,  in  passing  from  one  value  to  another, 
passes  through  every  intermediate  value  in  order,  then'the  vari- 
able is  CO  nt  1 1 1, 1  tons  ;  otherwise  it  is  discontinuous. 

E.g.,  time  is  a  continuous  quantity,  ever  increasing  by  a  steady 
growtli,  and  the  time  of  day,  expressed  in  hours  and  parts  of 
an  hour,  is  a  continuous  number ;  but  if  entire  hours  only  be 
counted  and  the  fractions  rejected,  the  number  is  discontinuous. 


158  INCOMMENSURABLES,   LIMITS,   ETC.  [VII. 

If  a  function  of  a  continuous  variable  remain  real  and  finite 
as  long  as  the  variable  is  real  and  finite,  if  it  can  take  but  one 
value,  or  a  limited  number  of  values,  for  any  one  value  of  the 
variable,  and  if,  in  passing  from  one  value  to  another,  it  passes 
through  every  intermediate  value  —  such  a  function  is  a  con- 
tinuous function;  otherwise  it  is  discontinuous.  It  is  implied 
that  for  any  small  increment  of  the  variable  the  increment  of 
the  function  is  also  small,  and  that  to  the  variable  an  increment 
can  always  be  given  so  small  that  the  increment  of  the  function 
shall  be  smaller  than  any  assigned  number. 

E.g.,  interest  is  earned  continuously,  and  may  be  computed  for 
a  3'ear,  a  day,  a  minute,  a  second,  a  millionth  part  of  a  second,  or 
any  other  fraction  of  a  second,  however  small ;  interest  is,  there- 
fore, a  continuous  function  of  the  time.  But  in  ordinar3'  busi- 
ness fractions  of  a  day  are  neglected,  and  interest,  having  definite 
sensible  increments,  is  a  discontinuous  function  of  the  time. 

So,  with  a  falling  bod}',  the  force  of  gravit}'  is  constant,  but 
the  time,  velocity  acquired,  and  distance  fallen  are  variables  ;  and 
the  velocity  and  distance  are  continuous  functions,  of  the  time. 

So,  the  area  of  a  regular  pol3'gon  inscribed  in  a  given  circle  is 
a  function  of  the  number  of  sides,  and  varies  with  the  number 
of  sides  ;  but  neither  the  number  of  sides  nor  the  area  is  a  con- 
tinuous number.  For  while  there  may  be  regular  poh  gons  with 
3,  4,  0,  •••  or  an}'  integral  number  of  sides,  it  is  absurd  to  speak 
of  such  a  pol3'gon  of  3 J  sides,  4f  sides,    and  so  on. 

So,  the  approximate  value  of  the  fraction  ^,  expressed  b}'  the 
decimals  .3,  .33,  .333,  •••  is  a  function  of  the  number  of  3's 
emplo3'ed,  but  that  number  is  discontinous  and  so  is  the  value. 

So,  the  convergents  of  a  continued  fraction  are  functions  of 
the  partial  numerators  and  denominators,  but  not  continuous. 

So,  in  the  equation  4a^  —  9/  =  36,  2/  =  ±  i^{Aa^-  36) ,  and 
for  all  values  of  a;  <  "3  and  >'^3,  y  is  a  continuous  function, 
but  for  all  values  of  x  from  ~3  to  +3,  2/  is  discontinuous. 

So,  if  2/  =  1  :  ic,  y  is  a.  continuous  function  for  all  values  of  x 
except  x  =  0,  where  y  leaps  from  a  ver}-  large  negative  to  a  very 
large  positive  value. 


§  2.]  INCOlNtMENSURABLES.  159 


§  2.     INCOMMENSURABLES. 

If,  in  any  operation  upon  numbers,  the  result  cannot  be 
expressed  as  a  commensurable  number,  either  an  integer  or 
a  fraction,  but  commensurable  numbers  can  be  found  both 
greater  and  less  than  the  true  result  that  approach  indefinitely 
near  to  it  and  to  each  other,  such  result  is  an  incommensurable 
number. 

E.g.^    the  square  root  of  2  is  an  incommensurable. 

(a)  It  is  not  an  integer. 

For       (0)2  =  0,   (±1)^=1,   (±2)2  =  4, 
and  (±3)2,   (±4)2,...  each  >  2. 

(b)  It  is  not  a  simple  fraction. 

tn 
For      if  possible,  let  ^2  =  — ,  a  simple  fraction  in  its  lowest 

terms ; 

then         2  =  —  a  simple  fraction  in  its  lowest  terms,  [III.  th.5,  cr.3 
n^ 

I.e.,  an  integer  is  equal  to  an  irreducible  fraction, 

which  is  absurd ; 

.-.  ^2  is  not  a  simple  fraction.  q.e.d. 

(c)  Commensurables^  both  greater  and  less  than  ^2,  can  be 
founds  that  shall  differ  from  it  by  less  than  any  assigned  number^ 
however  small. 

For     •.•  (±1)2=1,  and  (±2)2  =  4, 

.-.  ±1  <V2.  and  ±2  >  -^2, 

and  each  of  them  ~  ^2  <  1 . 

So      •••  (±1.4)2=1.96,  and  (±1.5)2=2.25, 

.-.  ±  1.4  <  ^2,  and  ±  1.5  >  V^' 

and  each  of  them  ~  ^2  <  .  1 . 

So      •.•  (±1.41)2=1.9881,  and  (±  1.42)2  =  2.0164, 

.♦.  ±  1.41  <  ■y/2,  and  ±  1.42  >  -^2, 

and  each  of  them  ^  ■yJ2  <  .01. 

So  indefinitely,  however  small  the  difference  assigned. 


160  INCOMMENSUKABLES,   LIMITS,   ETC.  [^U- 

So  the  square  roots  of  4,  9,  16,  25,  36,  49,  64,  and  81  are 
commensurables  ;  but  of  all  other  integers,  and  of 
most  fractions,  lying  between  1  and  100  they  are 
incommensurables  ;  and  so  of  other  roots. 

So  the  logarithms,  to  the  base  10,  of  10,  100,  and  1000 
are  commensurable  ;  but  of  all  other  integers  be- 
tween 1  and  10000  they  are  incommensurables. 

Incommensurable  numbers  often  represent  the  attempt  to  ex- 
press the  numerical  measure  of  a  quantity  in  terms  of  a  unit 
that  has  no  common  measure  with  it.  If  expressed  in  terms  of 
some  other  unit,  the  number  might  be  commensurable. 

E.g.^  the  diagonal  of  a  square  is  incommensurable  with  its 
side  ;  but  in  terms  of  the  half  diagonal,  or  any  other  exact  part 
of  itself,  say  /^ths,  ffds,  ffths,  •••,  it  is  commensurable. 

So,  time  may  be  expressed  in  days,  in  lunar  months,  or  in 
years  ;  but  it  is  very  unlikely  that  a  given  length  of  time,  exactly 
expressed  in  any  one  of  these  units,  would  be  commensurable  in 
either  of  the  others. 

So,  if  two  distances,  a b  and  c d,  be  taken  at  random, 

the  chances  are  few  that  ab  is  a  measure  of  cd,  or  that  they 
even  have  a  common  measure.  If  they  seem  to  have  one,  it  is 
probably  because  most  measurements  are  inexact,  and  only  rough 
approximations  are  used  instead  of  the  true  numbers,  which  are 
commonl}-  incommensurable. 

The  words  addition,  subtraction,  multiplication,  division,  and 
involution  to  commensurable  powers,  were  defined  in  I.  §§  6-11 ; 
and  those  definitions  were  made  so  broad  as  to  cover  all  kinds 
of  numbers.  The  axioms  laid  down  in  II.  §  3  also  apply  to  all 
numbers. 

Incommensurable  powers  and  logarithms  are  defined  in  VIII. 
§4,  IX.§1. 

The  combinatory  properties  of  commensurable  numbers  were 
proved  in  II.  §§  4,  6,  7  ;  the  same  properties  are  proved  for  in- 
commensurables in  VII.  §  7. 


§3.]  LIMITS.  161 


§  3.     LIMITS. 

When  a  variable  takes  successive  values  that  approach  nearer 
and  nearer  to  a  given  constant,  so  that  the  difference  between 
the  variable  and  the  constant  is  ver}^  small,  and  ma}'  become  and 
remain  smaller  than  an}-  number  named  or  conceived  of,  then  the 
constant  is  the  limit  of  the  variable  ;  and  this  definition  applies 
whether  the  variable  be  always  greater,  or  alwa3's  less,  or  some- 
times greater  and  sometimes  less,  than  the  constant. 
E.g.,    1,  1.4,  1.41,  1.414,  1.4142,  ... 

are  successive  approximations  to  the  true  value  of  y2, 
and  if  the  series  be  extended,  a  succession  of  terms  may  be 

found  whose  differences  from  y'2  are  smaller  than 
any  assigned  nmnbci'.   juid  steadily  grow  smaller 
and  smaller  as  the  series  goes  on,  but  which  terms 
are  each  less  than  -^2, 
So        2,   1.5,  1.42,   1.415,   1.4143,  ... 

are  each  greater  than  -y^2,  but  approach  it  nearer  and 
nearer  without  end ; 
.*.  while  -y/2  can  never  be  exactl}"  expressed  in  decimals, 
it  is  the  limit  to  which  both  the  series  approach. 

So        as  shown  under  continued  fractions,  ^2  =  1  +; —  1 

and  the  successive  convergents,  2 -!-••• 

3     7     17     41     99     239 

2'    5'    12'    29'    70'    169'  **' 

are  alternately  greater  and  less  than  -y/2,  the  true 

value,  but  approach  it  nearer  and  nearer  as  a  limit. 

7i  ±  1 

So        1  is  the  limit  when  n  increases  without  bounds. 

ri 

For     •.•  ^^^  =  1  ±  \    and  1,  =1±1~1,  =0  when  n=oo, 
n  n  n  n 

wherein    "  =  0  "  =  grows  smaller  and  smaller  and  approaches  0 

as  its  limit, 

and  "  =  go"  =  grows  larger  and  larger  without  bounds  ; 

.'.  lim =  1,  when  n  =  go.  q.e. d. 


162  INC0:>OIEXSURABLES,  LIINHTS,   ETC.  [VII. 

So        i  is  the  limit  of  tho  series  .3,  .33,  .333,  .3333,  •••. 
So        if  from  the   series   ^  {i)\  HY,  {¥)\  -  (-^)"  a  new 
series  of  sums  be  formed  b}'  taking 

then         the  limit  of  this  series,  when  7i  =  oo,  is  1 . 

For       8i~l=i,.S2^1=i  =  (|)S  .93^1  =i  =  (^)^..., 
and  <?„  ~  1  =  (4^)"  ==  0,   when  n  =  oo.  q.  e.  d. 

So,  if  a  regular  polygon  be  inscribed  in  a  circle,  and  another 
be  circumscribed  about  it,  and  if  the  number  of  their  sides  be 
doubled  again  and  again,  the  area  of  the  circle  is  the  limit  of  the 
areas  of  both  the  polygons,  and  the  circumference  of  the  circle 
is  the  limit  of  their  perimeters.  The  two  areas  approach  nearer 
and  nearer  to  the  area  of  the  circle  and  to  each  other ;  but  one 
is  always  a  little  greater  and  the  other  a  little  less  than  the 
circle  ;  and  so  of  the  perimeters. 

So,  the  surface  and  volume  of  a  cone  are  the  limits  respectively 
of  the  surface  and  volume  of  an  inscribed,  and  of  a  circumscribed, 
P3Tamid  ;  the  surface  and  volume  of  a  c^'linder  are  the  limits  of 
the  surface  and  volume  of  an  inscribed,  and  a  circumscribed, 
prism,    and  so  on. 

In  these  examples,  as  in  all  others,  the  constants  -^2,  1,  -J,  ••• 
are  limits,  not  simply-  because  the  successive  values  of  the  vari- 
ables approach  nearer  and  nearer  to  them,  for  they  approach 
nearer  and  nearer  to  man}'  other  numbers  not  their  limits. 

E.g.^  the  series  1,  1.4,  1.41,  1.414,  1.4142,  •••  approaches 
nearer  and  nearer  to  10000,  which  is  not  its  limit  at  all. 

So,  the  area  of  the  inscribed  poh'gon  approaches  nearer  and 
nearer  to  the  area  of  the  circumscribed  square,  not  its  limit. 

The  constants  are  limits  because,  as  the  series  is  extended, 
some  one  of  its  terms,  and  all  the  terms  that  follow  it,  will  differ 
from  the  constant  by  a  number  smaller  than  any  assigned  num- 
ber, be  that  number  never  so  small ;  and  further,  because,  how- 
ever far  the  series  is  extended,  there  is  no  point  bej'ond  which 
its  successive  terms  are  each  of  them  equal  to  the  limit. 


§4.]  INFINITESIMALS   AND  INFINITES.  163 

§  4.     INFINITESIMALS   AND    INFINITES. 

A  VARIABLE  is  infinitesimal  if  it  can  take  values  smaller  than 
any  assignable  magnitude,  infinite  if  it  can  take  values  larger 
than  any  assignable  magnitude,  finite  if  neither  infinite  nor 
infinitesimal.  All  constants  except  0  are  finite.  Strictly,  the 
word  "  infinitesimal"  applies  only  to  that  part  of  the  series  of 
values  of  the  variable  which  are  smaller  than  any  number  that  can 
be  named  or  conceived  of,  and  "  infinite"  to  that  part  of  the  series 
of  values  which  are  larger  than  can  be  named  or  conceived  of. 

Manifestly,  the  difference  between  a  variable  and  its  limit  is 
an  infinitesimal. 

The  reader  must  carefully  note  the  distinction  between  an 
infinitesimal  and  absolute  nothing.  The  latter  comes  from  sub- 
tracting any  number  from  itself ;  the  former  from  dividing  any 
number  into  small  parts  and  then  continually  subdividing  one  of 
these  parts.  An  infinitesimal  always  has  some  magnitude,  but 
absolute  nothing  means  the  total  absence  of  anything  to  measure. 

So,  between  the  infinites  of  mathematics  and  the  absolute 
infinity  of  space  and  duration,  there  is  the  same  impassable  gulf. 
Absolute  infinity  means  that  boundlessness  to  which  nothing  can 
be  added,  and  from  which  nothing  can  be  taken  away,  and  there 
are  no  means  by  which  it  can  be  measured  ;  but  a  mathematical 
infinite  is  simply  "  a  number  larger  than  can  be  named  or  con- 
ceived of,"  and  one  such  infinite  may  be  larger  than  another,  or 
any  number  of  times  another. 

The  essential  properties  of  infinitesimals  and  infinites,  upon 

which  the  mathematician  rests,  are  :  that,  while  following  the  law 

by  which  successive  values  are  determined,  the  one  may  be  made 

smaller  and  smaller,  and  the  other  larger  and  larger,  at  pleasure. 

T  •       4.  I  infinitesimals  ^  ,, 

In  comparing  two  or  more  ■{  .   n   .,  ^  any  one  of  them 

may  be  assumed  at  pleasure  as  the  base;  and  if  the  limiting  ratio 
(limit  of  ratio)  of  any  other  of  them  to  the  base  be  finite,  then 
that  other  number  is  of  the  same  order  as  the  base.     If  this 

limiting  ratio  be^     '  then  that  other  number  is^  \n^nite  ^^'^ 


164 


INCOMMENSUEABLES,   LEMITS,   ETC. 


[VII. 


as  to  the  assumed  base.    If  the  limitino:  ratio  of  aii<!  |"fi"|tesimal 

^  '  lulmite 

to  the  ?ith  power  of  the  base  (whatever  n)  be  a  finite  number, 

then  the  ^  j^fl^^g  ^^^  is  of  the  nth  orde)-  as  to  the  base,  and 

.         ,  infinitesimals  „   .,  1-^^.11  ^   •. 

two  <  .  r.  ..  are  01   the  same  order  11   they  have  nmte 

'  iniinites  "^ 

limiting  ratios  to  tlie  same  power  of  the  base. 

E.g.,  if  upon  any  Ntiaiuht  line  ab 
a  semicircle  be  doscriluMl.  niul  from 
c,  an}'  point  of  the  eiicu inference, 
CD  be  drawn  perpendicular  to  ab, 
and  AC  and  cb  be  joined, 
then         ABC  and  cbd  are  similar 

right  triangles, 
and  ab  :  bc  =  bc  :  db. 

Let  c  move  towards  b, 
then         AB  is  consiMut  Mlid  bc  and  db  are  variables. 

Let  c  approach  indefinitely  near  to  b, 
then  BC  is  an  infinitesimal  of  the  first  order, 

and  DB  of  the  second  order,  as  to  bc. 

For       •.•    AB  •  DB  =  BC^, 

.-.  lim  (bg-  :  db)  =  ab  a  finite  length. 

So,  if  in  the  triangle  abc, 
right-angled  at  c,  perpendicu- 
lars be  let  fall  from  c  on  ab  at 
D,  from  D  on  bc  at  e,  from  e 
on  db  at  F,  from  f  on  be  at  g, 
and  so  on  ; 
then  the  triangles  ABC,  cbd, 

dbe,  ebf,  fbg,  ••• 

are  all  similar, 

and      BC  :  AB  =  DB  :  BC  =  BE  :  DB  =  FB  :  BE  =  BG  :  FB  =  •  •  • . 

Conceive  the  point  c  to  move  towards  b,  and  to  approach 
indefinitely  near  to  it,  then  the  ratios  grow  smaller  and  smaller, 
and  finally  become  infinitesimals,  and  the  lengths  db,  be,  fb, 
BG,  •••  are  infinitesimals  of  the  1st,  2d,  3d,  4th,  •••  orders  as  to 
the  constant  length  ab. 


[above 
Q.E.D.      [df. 


§  5.]  DERIVATIVES.  165 

§  5.     DERIVATIVES. 

If  to  a  variable  a  small  increment  be  given,  and  if  the  corres- 
ponding increment  of  a  function  of  the  variable  be  determined, 
then  the  limit  of  the  ratio  of  the  increment  of  the  function  to 
the  increment  of  the  variable,  when  the  increment  of  the  vari- 
able is  taken  indefinitel}'  small,  is  the  derivative  of  the  function 
as  to  the  variable. 

E.g.,  let  a  square  p3'ramid  be  cut  b}'  planes  parallel  to  the 
base ;  the  sections  are  squares,  and  they  grow  larger  as  the 
cutting  planes  recede  from  the  vertex. 

Take  the  sides  of  two  squares  6  inches  and  7  inches  ; 
then  (72  -  62)  :  (7  -  G)  =  13  :  1  =  13. 

Take  the  sides  of  two  square  6  inches  and  6.1  inches 
then  (6.12-62)  :  (6.1  -  6)  =  1.21  :  .1  =  12.1. 

Take  the  sides  of  two  squares  6  inches  and  6.01  inches, 
then  (6.012—62)  :  (6.01-6)  =  .1201  :  .01  =  12.01. 

Take  the  sides  of  two  squares  6  inches  and  6.001  inches, 
then  (6.0012  _  (52)  .  (^c,.001  -  6)  =  .012001  :  .001  =  12.001  ; 

It  thus  appears  that  as  the  difference  of  sides  grows  smaller, 
1,  .1,  .01,  .001,  ...towards  0, 
so  also     the  difference  of  areas  grows  smaller  indefinitely, 

13,  1.21,  .1201,  .012001,  ...  towards  0, 
but  that  the  ratio  of  these  differences,  though  growing  smaller, 
has  12  and  not  0  for  its  limit. 
13,  12.1,  12.01,  12.001,  ...  towards  12  ; 
i.e.,  just  as  the  side  of  the  square  reaches  and  passes  6  inches  in 
its  growth,  at  that  instant  the  area  is  growing  12  times  as  fast 
as  the  side  ;  as  it  reaches  and  passes  7  inches,  14  times  as  fast ; 
as  it  reaches  and  passes  8  inches,  16  times  as  fast,  and  so  on  ;  and, 
in  general,  as  it  reaches  and  passes  x  inches,  2x  times  as  fast. 

When  two  variables  grow  smaller  and  smaller  together,  their 
ratio  does  not  necessaril}',  nor  generally,  become  infinitesimal. 

E.g.,  if  a  be  any  number,  however  small,  and  mb,  nb  be 
smaller  than  a, 
then         mb  :nb  =  m:n,  whatever  m  and  n  may  be. 


166 


INCOMMENSURABLES,   LIMITS,   ETC. 


[VII. 


So 

let  «=!, 

1 
20' 

1 
400' 

1 

8000' 

1 

I6OOOO' 

and 

y=h 

1 
2' 

1 
4' 

1 
8' 

1 

16'   * 

then 

x:y  =  l, 

1 
10' 

1 
lOO' 

1 
1000 

1 

10000' 

and 

y:  x=l, 

10, 

100, 

1000, 

10000,  . 

So 

let.=  J^ 

2 
'2.3' 

2 

3":4' 

2   2 
4.5'  5.G' 

2   2 
6.7'  7.8'  * 

and 

2/=  1, 

1 

1 

1   1 

1   1 

F  7^'  * 

then 

x:y=   1, 

4 
3' 

0 
G 

16  25 
To'  l5' 

36  49 
21'  28' ' 

•  towards  0, 

•  towards  0 ; 

•  towards  0, 

•  towards  00. 

•  towards  0, 

•  towards  0 ; 
••  towards  2. 


If  ?/  be  a  function  of  «,  then  the  phrase  "  derivative  of  y  as 
to  x"  is  written  d,2^,  wherein  d  stands  for  "  derivative  of,"  and 
the  subscript  x  for  "  as  to  x."  This  phrase  is  read  more  briefly, 
''  the  X  derivative  of  y" 

So,  T)yX=  the  derivative  of  a?  as  to  y,  or  the  y  derivative  of  x. 

Manifestly,  DyO;  is  the  reciprocal  of  D^y^  i.e.,  D^?/-Dyic=  1, 
inc.  X    inc.  y 


for 


=  1,  however  nearly  the  fractions  have 


inc.  y     inc.  x 

come  to  their  limits  D^ic,  d^?/, 
;•.  T)^X'D,y=\, 

Under  the  general  heading  of  this  chapter  the  reader  will  find 
three  classes  of  problems,  and  the  theorems  that  follow  lay  the 
foundation  for  rules  for  their  solution  : 

1 .  Those  which  involve  the  limits  of  variables. 

2.  Those  which  involve  the  ratios  of  two  infinitesimals. 

3.  Those  w^hich  involve  the  sums  of  an  infinite  number  of 
infinitesimals. 

To  the  first  class  belong  the  various  examples  given  under 
the  head  of  incommensurables  and  limits  ;  to  the  second  class 
belong  those  under  the  head  of  derivatives ;  and  to  the  third, 
the  computation  of  areas  and  volumes,  with  other  like  problems. 
The  process  last  named  is  called  integration. 


§6.]  FIRST   PRINCIPLES.  167 

§6.     FIRST  PRINCIPLES. 

Theor.  I.  If  two  variables^  the  one  increasing  and  the  other 
decreasing^  approach  each  other  so  that  their  difference  becomes 
infinitesimal,  they  have  a  common  limit  that  lies  between  them. 

1.  Each  of  them  has  some  Hmit. 

For,  if  either  had  no  Hmit,  they  would  pass  each  other. 

2.  The}^  have  the  same  limit. 

For,  if  they  had  different  Kmits  they  could  come  no  nearer 
together  than  their  limits.  [§  3  df.  lim. 

3.  This  common  limit  lies  between  the  two  variables. 

For  it  is  greater  than  the  less,  and  less  than  the  greater  of  them. 

Cor.  If  two  constants  always  lie  between  tiuo  such  variables, 
they  are  equal  to  the  common  limit  and  to  each  other. 

For,  if  possible,  let  one  of  them  be  greater  than  the  limit ; 
then         the  greater  variable  can  get  no  nearer  the  limit  than 
this  constant,  which  is  absurd.  [§  3  df.  lim. 

So,       neither  of  the  constants  can  be  less  than  the  limit. 
.*.  they  are  equal  to  the  limit,  and  to  each  other.      q.e.d. 

Theor.  2.  The  product  of  a  finite  number  into  an  infinitesimal 
is  an  infinitesimal,  and  of  the  same  order. 

Let  n  be  any  finite  number,  and  a  an  infinitesimal ; 
then  will  n-a  be  an  infinitesimal,  and  of  the  same  order  as  a. 

1 .  ?i  •  a  is  an  infinitesimal. 

For,  take  y8  any  finite  number  however  small,  and  a  <  /3  :  n  ; 
then         72  •  a  <  /?,  and  is  inf  1.      q.e.d.    [§  4  df .  infl.  II.  ax.  16 

2.  n- a  is  an  infinitesimal  of  the  same  order  as  a. 

For  71  •  a  :  a,  =  n,  is  finite.  q.e.d.     [df .  infl.  of  same  ord. 

Cor.  1 .  The  sum  of  a  finite  number,  n,  of  infinitesimals  is  inf^l. 

For  their  sum  <  n  times  the  largest  of  them,  [II.  ax.  12 

and     •.•  that  product  is  infinitesimal,  [th. 

.*.  the  sum  is  infinitesimal.  q.e.d. 

Cor.  2.    If  there  be  any  finite  number  of  commensurable  vari- 
ables, x',  y',  z',  •••,  and  as  many  more,  x",  y",  z",  ••♦,  such  that 
i'^x"  =  0,  y'~y"  =  0,   z'~z"i:0,...; 

then  will  x' -f  y' -fz' -h -.•  ~  x"+y"  +  z"H =  0, 

and  x'  •  y'  •  z'  •   •••  ~  x"  •  y"  •  z"  +  .-.  =  0 . 


168  INCOMMENSURABLES,   LIMITS,   ETC.  [VII. 

§  7.     PRIMAEY   OPElSriONS   ON   INCOMMENSURABLES. 

Theor.  3.     The  addition  of  incommensurahles  is  commutative 
and  associative. 

For,  let  a,  6,  c,  •••  be  any  incommensurables, 
I  x\  x"  ,  x'  <a  <  x'\ 

and  IPt  J  y'^  y"  ^^  commensurable   \y'  <b<  y'\  p.  ^  /if 

^^^  ^^M  z',  z"  variables  such  that  ]  z'  <  c  <  z",  LS  ^  «*. 

and  such  that  a  is  the  limit  of  «'  and  »",    6  of  y'  and  2/'\ 

c  of  z'  and  z",  •••  ; 
then   •.•  cc'-hy'+z' +...<«  4-&  +  c  +  -"<  a;"+2/"H-z"+..., 

[II.  ax.  12 

and  z'  -f  ?/'  -h  .^•'  +  •••  <  c  +  6  +  a  +  ...  <  z"+  y"+  x"+  ..-, 

and     •.•  a;' -f?/' +z'  + ..-,   =  z' -{- y' -{- x'  +  •••,  [Il.th.l 

=  a;"-h!/"4-z"+...,   =z"H-2/"+a;"  +  ..-,  [tli.2cr.2 

.*.  the  constant  sums  a  +  6+  c  H ,  c  +  ft+a  H ,  l3'ing 

between  these  two  variable  sums,  are  equal,  [th.  1 
So        for  any  other  order  or  grouping  of  the  elements  in  the 
sum  of  a,  6,  c,  •".  q.e.d. 

Theor.  4.     The  multiplication  of  incommensurahles  is  commu- 
tative and  associative. 

For,  let  a,  &,  c,  •••  be  an}'  incommensurables, 
I  a',  x"  I  a?'  <  a  <  x", 

and  let     y','  K  ^^  ?0";'"«"«"/-<lWe  \  y'<^<  v";  [§  2  df. 

I  z',  z"  variables  such  that  |  z'  '^  c  ^  z",  "-^ 

*">  *"  ■**  ***? 

and  such  that  a  is  the  limit  of  x'  and  a;",    b  of  ?/'  and  ?/"? 

c  of  z'  and  z",  .••  ; 
then   •••  x'  -y'  -  z' -"  ^a-b-c-"  <  a;".?/'',  z"--.,        [II.  ax.  19 
and  z'  -y'  -x'  -"  ^cb-a-"  <  z"  'y"'x"'", 

and     •.•  a:'  .?/'  .  z'...,   =  z'  .  ?/' .  a;' ...,  [II.th.3 

=  a;"..v".z"...,  =  z".2/".a;"...,  [th.  2  cr.  2 

.  • .  the  constant  products  a'b'C"-,  c-b-a"-,  l3'ing  between 
these  two  variable  products,  are  equal,    [th.  1  cr.  1 
So        for  an}-  other  order  or  grouping  of  the  elements  in  the 
product  of  a,  6,  c,  •«•.  q.  e.  d. 


§8.]  GENERAL   PKOPERTIES    OF   LIMITS.  169 

Theor.  5.     The  multiplication  of  incommensurables  is  distribu- 
tive as  to  addition. 

The  proof  is  identical  with  that  of  [II.  th.4.]. 

§  8.     GENEEAL    PROPERTIES    OF    LIMITS. 

Theor.  6.     If  two  variables  be  ahvays  equal,  and  if  one  of 
them  approach  a  limit,  the  other  approaches  the  same  limit. 

For,  let  X,  y  be  two  variables,  always  equal,  and  a  the  limit  ofx\ 

then   •.'  xr^a^O,  [§  3  df. 

and    • .  •  y  =  x     alwaj^s,  [hyp. 

.-.  2/~«  =  0;  [I.§5df. 

I.e.,  a  is  the  limit  of  y.  q.  e.  d. 

Cor.     If  while   approaching  their  limits,  two  variables  be 
always  equal,  their  limits  are  equal. 

Note.     Another  and  independent  demonstration  of  this  corol- 
lary is  as  follows : 

Let  X,  y  be  two  variables,  always  equal,  and  a,  b  their  limits, 
then  will  a  =  b. 

For      if  not,  let  a  ~  6  =  S, 
then   *.•  a,b  are  both  constantsi, 

.-.  8  is  a  constant,  however  small  it  may  be. 

Take    x,  y  such  that  X'^a  <i8,  and  y^^bK^S,  [df. 

then   •••  a;  =;?/ always,  [hyp. 

.-.  a^J)<8,  [11.  ax.  12 

which       is  contrary  to  the  supposition  that  a  '^  6  =  8  ; 

.*.  that  supposition  fails, 
and  it  is  only  left  that  a  =  b.  q.e.d. 

Theor.  7.     If  there  be  any  finite  number  of  variables  having 
limits,  the  sum  of  their  limits  is  the  limit  of  their  sum. 

Let  X,  y,  z,'"  be  any  finite  number  of  variables,  and  a,  &,  c,  ••• 

their  limits  ;  then  is  the  sum  a-\-b-\-c-\ the  limit  of  the  sum 

x->ry-{-z+  '". 

For     •.'  x  =  a-{-a,   y  =  b-\-p,   2;  =  c  +  y,  •••, 
wherein   a,  (3,  y,  -"  may  be  positive  or  negative,  but  each  of 
them  =  0, 


170  INCOMMENSURABLES,   LIMITS,   ETC.  [VII. 

...  x-hy+z  +  -"=(a-^a)-{-(b-{-l3)-{-(c-\-y)-\.-'  [II.ax.2 

=  («+^+c+-)  +  («+/?+y+-);  [tii.3 

and     •.•  a+^+7  +  ---=0,  [th.2 

.-.  x-\-y+z-\ =  a+b-\-c-\ asitslimit.    q.e.d.  [§  2df. 

Note.  When  the  number  of  terms  is  infinite  this  tlieorem 
does  not  alwaj's  appl}'. 

E.g.,    if  a,  a  finite  number,  be  divided  into  x  parts, 

then         limf -H--H-5h —  toa;terms  )isawhena;  =  ooand- =0. 
\x    ^    ^  )  ^ 

Theor.  8.  If  there  he  any  finite  number  of  variables  having 
limits,  the  limit  of  their  product  is  the  ^woduct  of  their  limits. 

Let  a;,  y,  z,  •••  be  any  variables,  a,  6,  c,  •••  their  limits  ;  then  is 
the  product  a-b'C'"  the  limit  of  the  product  x-y-z  "'. 
For     •.•  a;  =  aH-a,   y  =  b-\-p,   z  =  c-\-y,  ..., 
wherein   a,  )8,  y,  •••  may  be  positive  or  negative,  but  each  of 
them  =  0, 
.-.  x-y-z  •"=(a  +  a)'{b+li)'{c-hy)  '"  [II.ax.4 

=  a'b'C  — f-,  a  finite  number  of  terms,  each 
of  which  has  one  or  more  of  the  factors 
a,  )S,  y,  •••,  and  is  therefore  an  infini- 
tesimal ;  [ths.  5,  3 
and    •  .*  the  sum  of  finite  multiples  of  a,  y8,  y,  •••  =  0,  [th.  2 
.*.  a;'?/-2-«' ==a-6«c  •••  as  its  limit.                  q.e.d.     [df. 
CoR.  1.     If  there  be  two  variables^  the  quotient  of  their  limits 
is  the  limit  of  their  quotient. 

Let  a;,  y  be  an}-  two  variables,  and  a,  b  their  limits ;  then  is 
the  quotient  a  :  6  the  limit  of  the  quotient  x :  y. 
For  let     x^y-q,  wherein  q  is  the  quotient  of  x  by  y, 
then   *.•  a  =  6-lim^,  [th. 

.-.  a:  6  =  limg,  q.e.d. 

Cob.  2.  Any  finite  integral  power  of  the  limit  of  a  variable  is 
the  limit  of  the  like  j^oiuer  of  the  variable. 

Let  X  be  anj^  variable,  a  its  limit,  n  an}^ integer,  then  a" = lima;", 
(a)  n  positive :  a  case  of  multiplication. 

(6)  n  negative :  a  case  of  division. 


§  8.]  GENERAL  PEOPEP.TIES   OF  LII^nTS.  171 

Note.  When  the  exponent  is  infinite,  this  corollary  does  not 
always  apply. 

E.g.,    (l-f--Y  whena;  =  oo,  is  not  1  but  2.718H ,     [ 

CoR.  3.   If  there  be  two  variables  x,  y  and  two  others  x',  y', 
such  that  lim  (x :  x')  =  1  and  lim  (y  :  y')  =  1, 
then         lim  (x  :  y)  =  lim  (x' :  y') . 

Theor.  9.  If  there  be  two  variables  x,  y  whose  limiting  ratio 
is  a  finite  number^  not  0,  and  if  there  be  added  to  them  any 
numbers  a,  /8,  infinitesimal  as  to  x,  y,  then  is  the  limiting  ratio 
of  X,  y  not  changed. 

For    •  •  ^il^  =  ^.^+(^-^), 
2/  +  )8      y     l+((3:y) 

...  lini£±^=lim^  .  lim^  +  ("^^)^lim^  .  l  =  lim^. 
?/  +  /3  y  l  +  {/3:y)  y     1  y 

Q.E.D. 

CoR.  If  the  difference,  S,  of  two  variables  x,  y  be  inflriitesimal 
as  to  either,  their  limiting  ratio  is  1 ,  and  conversely. 

Theor.  10.  If  x,  3',  z,  be  three  variables  of  the  same  sense, 
such  thai  x  <  3'  <  z,  and  such  that  x  :  z  ==  1,  then  will  x  :  y  =  1, 
and  3" :  z  =  1. 

For    •.•  x^y^z,  [hyp. 

.*.  X  :  z<^x  :  y<^y  :  y,  i.e.,  <  1,  [II.  ax.  18 

and  X  :  z  -^  y  :  z  '^  z  :  z,    i.e.,  -^1,  [II.  ax.  17 

But    • .  •  X  :  z  =  1,  [tiyP- 

.  • .  X  :  y,  ^  X  :  z  but  <  1 ,  =  1 , 
and  y  '  z,  ^  X  :  z  but  ^  1,  ==  1.  q.e.d. 

Theor.  11.     Ifx,ybe  two  infinitesimals,  m,  n  their  orders  as 

to  any  base  /8,  and  if  m  >  n,  then  is  (x  :  y)  an  infinitesimal  of 

the  (m  — n)^/i  order  as  to  the  base. 

For    • .  •  lim  (a;  :  ^'")  =  h  and  lim  {y  :  /3")  =  A;, 

wherein    h  and  k  are  finite  numbers,  [^yP* 

lim  (x  :  y)         lim  (x  :  B"^)         /i        /?   ..  u 

. • .     ,.  ^  ^    '^^ ,  =  - — ) ^,   =  -,  a  finite  number 

hm^"'-"  \im(y  :  (S"")         k 

.  • ,  (x  :  y)  isan  infinitesimal  of  the  (m  --?i)  th  order,  q.  e.  d. 


172  INCOMMENSURABLES,   LIMITS,   ETC.  [VII. 

Cor.  1.  The  product  X'j  is  an  infinitesimal  of  the  (m  -j-  \'\)tli 
order. 

Cor.  2.     3'  :  x  is  an  infinite  of  the  (m  —  i\)th  order. 

Cor.  3.  If  x,  y  he  infinites  of  the  mth  and  nth  orders,  and  if 
m  >  n,  then : 

X  :  3'  is  an  infinite  of  the  (m  —  n)th  order, 
X  •  y  is  an  infinite  of  the  (m  +  xi)th  order, 
y  :  X  is  an  infinitesimal  of  the  (m  —  n)th  order. 
If  there  be  two  or  more  numbers  not  all  equal,  then  any  num- 
ber which  is  greater  than  the  least  of  them  and  less  the  greatest 
is  a  mean.    The  average  of  two  or  more  numbers  is  the  quotient 
of  their  sum  by  their  number. 

Theor.  12.  If  x',  x",  x'",  •••  be  a  set  of  variables,  and 
y\  y"?  }"'"•>  '"CIS  many  more,  all  positive  or  all  negative,  and 
Inch  that  lim  (x' :  y')  =  1 ,  lim  (x"  :  y ")  =  1 ,  lim  (x'"  :  y'")  =  1 ,  •  •  • ; 
and  if. the  number  of  variables  in  each  set  increase  without  bounds, 
then  the  limits  of  the  sums  of  the  two  sets,  unless  infinite,  are  equal. 

For    •.•  lim(a;':y)=l,  lim(a;":  2/")  =  l,  lim(aj"' :  2/'")  =  l,  ..., 


2/' +  2/" +  2/'"+-         ' 
.    lim(x^+a;^^+a?^^^+-)^l 

lim(2/'4-2/"+2/"'+"-) 
.-.  lim  (a;'+ «"+«"'+. ..)  =  lim(2/'  +  2/"4-2/"^+---)-Q-E.D. 

CoR.  i/'lim(x' :  y')  =  m,  lim(x"  :  y")  =  m,  •..,  andifx.',  x",  ... 
y'j  y"?  •••  ^e  all  positive  or  all  negative,  then 

lim[(x'+x"+...):(y'+y"+-)]  =  ni. 

Note.  This  theorem  is  of  great  service  in  geometry  in  com- 
puting areas,  volumes,  etc.,  bounded  by  curved  lines  or  surfaces. 
Divide  into  narrow  bands  whose  limits  are  rectangles,  or  thin 
plates  whose  limits  are  prisms,  and  then  get  the  limit  of  the 
sum  of  such  rectangles  or  prisms ;  these  limits  are  the  areas  ot 
volumes  sought.     This  operation  is  called  integration. 


1.  §  9.]        GENERAL  PROPEETIES   OF  DERIVATIVES.  173 

§  9.     GENERAL  PEOPERTIES  OF  DERIVATIVES. 

PrOB.  1.  To  FIND  THE  DERIVATIVE,  AS  TO  ANY  VARIABLE,  OF 
A    FUNCTION    OF   THAT  VARIABLE. 

In  the  function  give  the  variable  an  increment;  from  the  re- 
sulting expression  subtract  the  function;  divide  the  remainder  by 
the  increment  of  the  variable^  and  get  the  limit  of  the  quotient  as 
tfiat  increment  approaches  zero. 

E.g. ,    to  find  the  derivative  of  x^ : 

Let  X  =  ar^ ;  substitute  x-{'h  for  oj,  and  let  x'  =  (a;  +  hy ; 
then   •••  x'-x       ={x  +  hy-x'  =  2xh  +  h\ 
x'— X 


=  2x  +  h, 


x'  — X 


.'.  lim =  2x     when7i  =  0, 

h 

i.e.,         i>x(^)        =2x.  Q.E.F. 

So        to  find  the  derivative  of  a^ : 

Let  X  =  ic' ;  substitute  a;  +  ^  for  x,  and  let  x'=  (a;  +  hy ; 
then    •.•  x'-x        ={x-hhy-a:^  =  Sx'h-{-Sxh^+JiK 

.-.  ^—^       =3ar»4-3a;;i  +  A2, 
h 

...  lim^^:^=:3ar^    when/i  =  0, 
h 

i.e.,         i>x(^^)        =3ar^. 

So        to  find  the  derivative  of  x~^ : 

Let  X  =  a;-^ ;  substitute  a;  +  ^  for  x,  and  let  x'=  (a;  +  hy^ ; 

then   *.•  x'— X       =(a;  +  7i)~^— a;'^ 

^_1 1        ^      -h 

x-\-h     x  x(x-\-h) 

x'-x  -1 


h  x{x-{-hy 


...  iim^!-^  =  — i    when^  =  0, 
h  ar 

i.e.,         I>^  =-i.  Q.E.F. 


174  INCOMMEXSURABLES,   LIMITS,   ETC.  [VH.  th. 

Theor.  13.     The  derivative^  as  to  any  variable,  of  the  sum 
of  two  or  more  functions  of  that  variable,  is  the  sum  of  their 
derivatives. 
Let  u,  V-"  be  any  functions  of  a  variable  x,  and  x  their  sum  ; 

then  will  d^x  =  d,u  -f  d^v  H 

For  let  X  take  any  infinitesimal  increment  /i,  and  let  x'  stand 
for  the  new  value  of  a?,  x'  for  the  corresponding  value 
of  X,  u'  for  that  of  u,  v'  for  that  of  v,  •••,  so  that 
x'=x-^h,  x'=x  +  incx,  Tj'=u  +  incu,  v'=v  +  incv, 

then   •••  X  =u+vH always,  [^^yP* 

.-.  x'  =u'+v'+— , 

.-.  x'— X     =u'— u+v'-v+--,  [II.ax.3 

i.e.,  incx        =incu    +incv    -\ , 


mqx        _incu     .  mcv 
h  h  h 


+  ...,  [n.  ax.5 


...  hmlH^  =  lim'-5£H  +  iimiH£I+...  when/i  =  0,    [th.  7 
h  h  h 

i.e.,         D^x,        =D,(u  +  v4----)» 

=  D,U+D,VH .  Q.E.D. 

Theor.  14.  TJie  derivative,  as  to  any  variable,  of  the  product 
of  two  or  more  functions  of  that  variable,  is  the  sum  of  the  prod- 
nets  of  the  derivatives  of  the  several  factors  each  multiplied  by  all 
the  other  factors. 

Let  u,  V,  w,  •••  be  any  functions  of  a  variable  x,  and  x  their 
product; 

then  will  Da,x  =  v-w-"D,u4-u«W"-d,v  +  u-v-"D,w  +  *"' 
For  let  X  take  any  infinitesimal  increment  h,  and  a;'  be  the 
new  value  of  x,  so  that  x^=.x-\-h,  x'  =  x  +  incx,  •••, 
then   •.•  X  =u-v«w  •••  always,  [hyp- 

.*.  x'  =zu'-v'- w'-", 

i.e.,         x+incx  =  (u  +  incu)-(v  +  incv)-(w+incw)  ••• 

=  U'V«w f-'v^-W'-'incu+U'W-'-incvH — 

+  terms  with  two  or  more  infin'l  factors, 


15.  §  9.]       GENERAL  PROPERTIES   OF  DERIVATIVES.  175 

.*.  incx  =  v-w-"incu  +  u-w---incv4-u.v---iiicw+  ••• 
+  terms  with  two  or  more  infin'l  factors, 

h  h  h  h 

+  terms  with  one  or  more  infin'l  factors, 

.-.  D,x,  =D,(u.v.w-..), 

+  terms  that  vanish.  q.e.d.     [th.  7 

Cob.     In  particular^  the  derivative^  as  to  any  variable,  of  the 
product  of  two  functions  of  that  variable,  is  the  sum  of  the  prod- 
ucts of  the  derivatives  of  the  two  functions  each  multiplied  by  the 
other  function. 
I.e.,         D,(u-v)  =  u-D^v4-v.D^u. 

Note.     Theorem  14  may  be  written  in  the  form : 

D^(U.V.W--.)__D,U       D£V-       D^W    , 


U-V-W»"  U  V  w 

Theor.  15.  TJie  derivative,  as  to  any  variable,  of  a  fraction 
whose  terms  are  functions  of  that  variable,  is  a  fraction  whose 
numerator  is  the  product  of  the  denominator  into  the  derivative 
of  the  numerator  less  the  product  of  the  numerator  into  the  deriv- 
ative of  the  denominator,  and  whose  denominator  is  the  square 
of  the  given  denominator. 

Let  u,  V  be  any  functions  of  a  variable  x,  and  x  their  quotient ; 

thenwUlD.x^^-^-^"-^-^-^. 

v^ 

For  let  X  take  any  infinitesimal  increment  h,  and  x'  be  the 
new  value  of  x,  so  that  x'  =  x  -\-  h,  x'  =  x  +  inc  x, 
u'  =  u  +  incu,  v'  =  v-|-incv, 

then   •.*  x  =  -  always. 


v-f-incv 


176  INCOM]VIENSUEABLES,   LIMITS,   ETC.  [VII.  th 

u  +  incu     U 


incx 

V 

+  incv 

V 

_  V 

•  incu  — 

u-incv 

V-+V. 

mcv 

inc  X 

V 

incu 
h 

h 

h 

v^'  +  v 

•  mcv 

.^       u    ^VD,u-u^D,y  ^^^      [th.7,tli.8,cr.l 

V  v'' 

Cor.  1.     Ifvbe  constant  and  v  a  function  of  x^ 

D,u  =  0,     ana    d,-= — 2^  =  u«d,— 

V  V-  V 

Cor.  2.     If^he  constant  and  u  a  function  of  x^ 

^  ,         u      D^U      1 

D,v  =  0,     and    d,-  =  -^  =- •d^.u. 

V  V  V 

Note.     Theorem  15  may  be  written  in  the  form  : 

u  /      In      1  _L  1 

i>x-  =  D.(u--)  =  --D.u  +  u.D.-. 

Theor.  16.  The  derivative,  as  to  any  variable,  of  a  function 
of  a  function  of  that  variable,  is  the  product  of  the  derivatives  of 
the  immediate  functions  which  compose  it,  each  taken  as  to  the 
variable  on  which  the  immediate  function  depends. 

Let  u  be  an}^  immediate  fmiction  of  a  variable  x,  and  x  any 
immediate  function  of  u  ; 
then  will  d^x  =  d^x  •  d^jU. 

For  let  X  take  an  infinitesimal  increment  h,  then  u  and  x  will 
take  corresponding  increments ; 

,  incx  incx  incu  ttt  +i    q   ^„^  i    n 

and     •  •  =  : ; — ,  [11.  th.  3,  crs.  1,  7 

h  mc  u      h 

...   limlH2Z  =  lim^-H^^limlH£I,when7i  =  0,  [th.6,cr.,th.8 
h  mc  u  h 

i.e.,  D,X  =DuX^D,U.  Q.E.D. 


17.  §  9.]       GENERAL  PEOPERTIES   OF  DERIVATIVES.  177 

Theor.  17.  The  derivative,  as  to  any  variable,  of  a  com- 
mensurable power  of  that  variable,  is  the  product  of  the  given 
exponent  into  a  power  of  the  variable  whose  exponent  is  a  unit 
less  than  the  given  exponent. 

Let  x  be  any  variable,  and  n  any  commensurable  number; 
then  will  d^x^  =  nic""^. 

(a)   n  a  positive  integer  : 
For    •••a;'*      =  the  product  a;- a; -a;-.  •,  w-times  repeated, 

.-.  D^ a;"  =  a;"~^«D^a;+a;**"^«D^a;4-.«.,w-times repeated  [th.l4 
=  ?ia;""^-D^a;. 
But    •.*  v^x  =  1, 

.*.  D^a;'*  =  ?ia;**~\ 

(6)   n  a  positive  fraction,  E ;  p,  q  both  positive  integers: 

q 

p 

For  let    X  =  »«', 


.*• 

5x'~^-D,x=j9a;^-^ 

[(a),th.l6 

••• 

1 

{xi) 

{x^y 

3CP-1) 
P(2-1) 

V       1 

p-q 
X    9 

p 

.H 

[11 

.  th. 

3,  crs.  10,  11 

i.e., 

D^a;**           zmnx""-^. 

Q.E.D. 

(c)  n 

any  negative  number,  —  m : 

For  let 

X      =  a;-*", 

then 

_-maf-i_                1 

[th.l5,cr.l 

X"              '""'        ' 

i.e., 

D^a;"  z=nx**~^. 

Q.E.D. 

Cor. 

If  V  be  any  function  of  x,  then 
D^u"  =  nu°"^-D,u. 

[th.  16 

178  INCOIkOIENSUKABLES,   LIMITS,   ETC.  [VII 

§  10.    INDETEEMINATE   FORMS. 

If  there  be  an  expression  that,  by  the  definitions  of  the  sj^m- 
bols  used,  may  take  an  infinite  number  of  different  values  lying 
in  a  continuous  series,  such  an  expression  is  indeterminate. 

[See  II.  §2,  p.  28. 

E.g.^    the  expressions  0:0,  oo  :  qo,  oo  — oo,  are  indeterminate. 

For      the  quotient  0  :  0  may  be  any  quotient  that,  multiplied  by 

or  into  the  divisor  0,  gives  the  dividend  0  as  product ; 

and  any  finite  quotient  may  do  this.  '-  '  ^        *    ^^* 

And     the  quotient  qo  :  oo  may  be  any  quotient  that,  multiplied  by 
or  into  the  divisor  oo,  gives  the  dividend  oo  as  product ; 
and  an}'  quotient,  not  0,  may  do  this. 

And     the  remainder  oo  —  oo  may  be  an}^  remainder  that,  added 
to  the  subtrahend  oo,  gives  the  minuend  oo  as  sum  ; 
and  any  remainder  may  do  this. 

So        the  quotient  xiyis  indeterminate  if  of  x,  y  it  be  known 
only  that  both  =  0,  or  that  both  =  oo. 

And     the  remainder  x  —  y\Q  indeterminate  if  of  cc,  y  it  be 
known  only  that  both  =  oo. 

For      any  number  may  be  such  a  quotient  or  remainder. 

If  for  a  particular  value  of  an^^  variable  of  which  its  terms 
are  functions  a  fraction  take  the  form  0:0,  it  ma}'  be  regarded 
as  approaching  this  form  by  gradual  change  of  the  variable,  and 
its  true  value  is  strictly  the  limit  of  the  ratio  of  two  infinitesimals. 

This  value  is  finite  when  the  terms  of  the  fraction  are  infini- 
tesimals of  the  same  order  [§§4,  5],  and  it  is  indeterminate 
only  so  long  as  the  law  is  unknown  subject  to  which  they  =  0. 

E.g.^    when  ic=  1,  a?—l :  a^— 1  becomes  0  :  0, 
but  when  a;  =  1  +  ^,  this  fraction  becomes 

(1  +  ^)2-1'  7i2-f27i  7i-t-2       ' 

=  3:2  when  h  =  0,    ^.e.,  when  a;  =  1, 
and  its  true  value,   when  a;  =  1,    is  3  :  2. 


§10.]  INDETERMINATE   FORMS.  179 

The  reader  will  see  that  this  process  is  equivalent  to  reducing 
the  given  fraction  to  its  lowest  terms,  then  substituting  1  for  x. 

In  general,  fractions  take  the  form  0  :  0  because  of  some  com- 
mon factor  of  their  terms  that  vanishes  for  a  particular  value  of 
the  variable.  If  this  factor  can  be  found  and  divided  out,  and 
the  particular  value  be  substituted  for  the  variable,  the  result  is 
the  true  value  of  the  given  fraction  ;  and  this  method  is  particu- 
lar 1}^  useful  for  fractions  whose  terms  are  entire. 

In  the  above  example  the  vanishing  factor  is  a;  —  1 ,  and  the 

x^  -\-x-\-  1      3 
fraction,  when  this  factor  is  divided  out,  becomes  — — — -^— -  =  — 

a?  +  l  2 

Theorem  18  will  show  another  method  of  evaluation. 

Expressions  that  approach  the  forms  go  :  oo,  oo  —  x,  may  be 

reduced  to  equivalent  expressions  that  approach  the  form  0 :  0. 

E.g.,  let  X,  x'  be  functions  of  any  same  variable  x,  such  that 

when  x  =  a,  then  also  x,  x'  both  =  oo. 

Put  X,  x'  under  the  forms  u :  v,  u' :  v', 

wherein   u,  u',  v,  y'  are  all  functions  of  x  such  that,  when  ic  =  a, 

V  is  an  infinitesimal  of  any  order,  and  u  is  finite  or  an 

infinitesimal  of  a  lower  order  than  v,       [th.ll,  cr.  2 

and  v'  is  an  infinitesimal  of  any  order,  and  u'  is  finite  or  an 

infinitesimal  of  a  lower  order  than  v', 

.1  f     .  u      u'       u«v'  — u'«v     .  0 

then         X  — x',  =oo— oo,  = ,  = ; — ,  =  — 

V      v'  V'V'  0 

E.g.,   if  u,  V,  u',  y'  =  a;  +  2,  x-1,  a^-1,  a^—2oe^-{-x, 

then   •.*  u  is  finite,  v,  u'  are  infinitesimals  of  the  first  order,  and 

y'  is  an  infinitesimal  of  the  second  order,  when  a;  =  1, 

.'.    X  — X',    =00  —  00, 

=  (x^'a^-2x^-\-x-x^l'X'-l):x(x-lf, 

:^0:0, 

=  {ij(P—x  —  l)  :x  —  l,  =00,  when  x  =  l. 

[div.  out  van.  fac.  (x—iy. 

It  has  been  shown  above  that  the  forms  called  indeterminate 

belong  to  that  class  of  limiting  expressions  wherein  the  variables 

cease  to  have  finite  values.     They  differ  from  other  limiting  ex- 


180  IXCOMMENSUKABLES,   LIMITS,   ETC.  [VII.  th. 

pressions  of  the  same  class  in  this,  that  their  limits  cannot  be 
determined  without  more  knowledge  of  the  relations  of  the  vari- 
ables than  appears  upon  the  face  of  the  expressions  themselves. 

E.g.,    when  ic,  y  both  =  0,  the  quotient  (S—x)  :  (4  —  ?/) ,  not 
an  indeterminate  form,    =  2,  no  matter  how  a;,  y 
may  be  related ; 
but  the  quotient  x:y^  =  0  ;  0,  ma}'  have  any  limit  whatever, 

depending  on  the  relations  of  the  variables  x,  y. 

From  this  point  of  view  the  form  oo  •  0  may  be  added  to  the 
list  of  indeterminate  forms ;  for  although  it  does  not,  like  the 
other  three  forms,  take  an  infinite  number  of  different  values  by 
the  mere  definition  of  the  symbols  taken  absolutely,  vet,  like 
them,  it  may  take  any  value  whatever,  considered  as  a  limiting 
expression,  i.e.  as  the  limit  of  the  product  of  an}^  two  variables, 
one  of  which  =  oo  and  the  other  =  0. 

An  expression  that  approaches  the  form  co  •  0  may  be  reduced 
to  an  equivalent  expression  that  approaches  the  form  0  :  0. 

Theor.  18.    If  for  a  particular  value  of  a  variable  two  func-^ 
tions  of  that  variable  both  vanish,  the  true  value  of  their  quotient 
is  the  quotient  of  the  values  of  their  derivatives  for  that  value  of 
the  variable. 

Let  x,  x'  be  two  functions  of  a  variable  x  such  that  x^,,  x '«, 
their  values  when  a  is  put  for  x,  both  vanish ; 
then  will  x„  :  x  „=  d^x^  :  d^x'^. 

For,  in  x,  x',  put  a  +  h  for  x  ; 


then 

*•*    Xfl+A                      — '^a  +  h         ^a?                                                                 L^a — ^ 

and 

X'a  +  .        ^           =XU*-X',,                            ^                                 [X'„=0 

•*•    Xa  +  A  :  Xa^;^  =  Xg  +  ft  — X^rXa^^ — X^ 

'^a-ith  —  '^a  .'^  a  +  h  —  '^  a 

h               '                h               ' 

.-.  lim(x„+»:x',+A) 

-  lim  (^-+»-  ^- :  ^'«+'i-  x'«)  ,when  h  -  0, 
h                  h 

-lim^''"'^~^«;lim^'''+'^~^'%[th.8,cr.l 
h                         h 

i.e., 

x«:x'«          =d,x„:d,x'„.                                q.e.d. 

T 

^2 

A3 

^1 

! 

1 
1 
1 

A4 

^2 

^3 

0 

Ai 

P4 

18.  §11.]  GRAPHICAL  EEPKESENTATION  OF  FUNCTIONS.      181 

§11.     GRAPHICAL  EEPEESENTATION   OF   FUNCTIONS. 

For  convenience  in  treating  of  integration  and  other  subjects 
discussed  later,  the  geometric,  words,  origin,  axis,  abscissa,  and 
ordinate,  are  here  defined,  and  the  reader  is  introduced  to  the 
method  of  representing  by  a  geometric  locus  an  algebraic 
equation  between  two  variables,  or  a  function  of  a  single 
variable. 

Let  p  be  any  point,  x'x  an}'  straight 
line  lying  in  the  plane  of  the  paper, 
and  o  a  fixed  point  on  x'x ;  from  p 
draw  PA  perpendicular  to  x'x  and  x^ 
meeting  it  at  a  ;  then  x'x  is  the  ref- 
erence  line,  or  axis,  the  fixed  point  o 
is  the  reference  point,  or  origin,  the 
line  OA  is  the  abscissa  of  the  point  P, 
AP  is  its  ordinate,  and  oa,  ap  together  are  the  coordinates  of  p. 

If  the  figure  lie  before  the  reader  so  that  x'x  is  a  horizontal 
line  with  x  to  the  right  of  x',  then  the  direction  x'x  is  ordinarily 
taken  as  the  positive  direction  and  xx'  as  the  negative  direction 
[I.  §  3]  ;  and  abscissas  measured  to  the  right  from  o  are  posi- 
tive, while  those  measured  to  the  left  are  negative.  So,  ordi- 
nates  measured  upward  from  the  axis  are  positive,  and  those 
measured  downward  are  negative. 

An  abscissa  is  generall}'  represented  by  the  letter  x,  and  an 
ordinate  b}^  y.  So,  the  line  x'x  is  called  the  axis  of  abscissas, 
or  the  axis  of  x;  and  the  line  y'y,  drawn  perpendicular  to  x'x 
through  o,  is  called  the  axis  of  ordinates,  or  the  axis  of  y. 

The  position  of  a  point  is  determined,  and  the  point  may  be 
constructed,  when  its  coordinates  are  given. 

When  the  coordinates  are  not  given,  but  are  connected  by  a 
given  relation  {e.g.,  that  their  sum  is  constant),  an  infinite  num- 
ber ^  of  points  may  be  found  that  satisfy  the  conditions,  for  if 
any  value  be  assumed  for  the  abscissa,  the  given  relation  between 
the  coordinates  serves  to  determine  the  corresponding  value  or 
values  of  the  ordinate. 


182  INCOIMMENSURABLES,   LIGHTS,   ETC.  [VII. 

In  general,  these  points  all  lie  in  some  line,  straight  or 
curved,  called  their  locus;  and  the  relation  between  the  variable 
coordinates  may  be  expressed  by  a  single  equation  between  two 
variables,  called  the  equation  of  ihe  locus.  In  this  equation 
either  variable  is  a  function  of  the  other.  The  equation  is  satis- 
fied by  the  coordinates  of  every  point  of  the  locus,  and  by  those 
of  no  other  point.  Such  equations  are  generally  written  in  the 
form  y=fx,  wherein  a;,  the  abscissa,  may  be  regarded  as  an 
independent  variable,  and  ?/,  the  ordinate,  as  a  function  of  x ; 
and  the  shape  of  the  locus  of  the  extremities  of  the  ordinates 
shows  the  manner  in  which  fx  varies  with  x. 

E.g.-,  the  locus  of  points  whose  coordinates  satisfy  the  rela- 
tion expressed  by  the  equation  y  =  mx  is  a  straight 

line  through  the  origin. 
V  Let   ox  be  the    axis,    o   the   origin,  ^ 

p,  p'  any  two  points  whose  co- 
ordinates OA,  AP,  oa',  A.'p'  are 

so  related  that  ap  =  m  •  oa,  and 

a'p'  =  m.  oa',  i.e.,  so  that  y  =  mx  for  each  of  them  ; 
then  is   pp'  a  straight  line  through  o. 

For    •.•  ap:oa  =  a'p':oa',  [hyp. 

and     •.•  AP  is  parallel  to  a'p',  [constr. 

.'.  the  straight  line  op  passes  through  p',  and  is  the  locus 

sought.  [geom. 

So,  the  locus  of  the  equation   y  =  mx-\-b  is  a  straight  line 

that  cuts  the  axis  of  y  at  a  distance  b  above  the 

origin. 
As  above,  construct  the  straight 

line    that    represents    the 

equation  y^^  mx  ;    draw 

any  two  ordinates  ap,  a'p', 

and  extend  them  to  q,  q', 

so  that  AQ  =  AP  +  6,  a'q'  =  a'p'  +  &,  wherein  h  is  any 

constant ; 
then  is  qq'  a  straight  line  parallel  to  opp',  and  the  locus  sought. 


§  11.]       GRAPHICAL  REPEESENTATION  OF  FUNCTIONS.       183 


So,  the  locus  of  the  equation  a^+?/-=r^, 

•wherein  r  is  constant,  is  a  circle 

whose   centre   is  the   origin    and 

whose  radius  is  r. 
The  reader  may  see  this  from  the  prin- 
ciple of  geometry  that  "  in  a  right  triangle 
the   square  of  the  hypotenuse   equals  the 
sum  of  the  squares  of  the  other  two  sides." 

So,  the  locus  of  the  equation   'if^px  is  a  parabola  whose 

axis  is  the  axis  of  a;,  whose  vertex  is  at  the  origin, 

and  whose  parameter  is  p  ; 
and  the  locus  of  the  equation  Q^-=.py  is  a  parabola  whose  axis 

is  the  axis  of  y. 
The  reader  will   y 
recognize     these 
equations  as  the 
algebraic  expres- 
sion of  the  geo- 
metric   property 
of  the  parabola, 
that  "the  square 
of  a  perpendicu- 
lar from  any  point 
of  the  curve  to  its  axis  equals  the  product  of  its  parameter  into 
that   part  of  the  axis   intercepted 
between  the  vertex  and  the  foot  of 
the  perpendicular." 

So,  the  locus   of  the    equation 

xy  =  (?  is  the  rectangular 

hyperbola,  taken  with  ref- 
erence to  its   asymptotes 

as  axes  of  coordinates. 
These  figures  also  represent  graph- 
ically  the   functions  mx^    mx  -f-  6, 


x" 
p^ 


— ,    and  show  how  they  vary  with  x, 

X 


184  DsCOM^IENSURABLES,   LIMITS,   ETC.  [VII.  t  . 

§  12.     INTEGRATION. 

Theor.  19.  If  there  he  a  variable  x,  and  if  fx  he  a  function  of 
X  whose  derivative  as  to  x  is  f  x  and  is  continuous;  and  if  the 
variable  begin  loith  the  value  Xq,  =  a,  and  take  n  more  successive 
values  Xi,  Xs,  •••Xn,  =b;  and  if  while  a  and  b  stand  fast. 
n  =  00  and  Xi  —  Xq,  Xg  —  Xj,  •••  each  =  0  ;  then  the  sum  of  the 
series  of  products  {xi-^Xo)f'xo,  (xj-xOf'xi,  •••(x^-x^_i)f'x„_i, 
approaches  fb  —  fa  as  its  limit. 

ILLUSTBATITB    EXAMPLES. 

That  the  reader  may  clearly  understand  the  meaning  of  the 
theorem  and  its  proof,  and  that  he  may  see  how  this  method  of 
summation  was  first  suggested,  and  follow  the  historical  order 
of  investigation,  special  applications  of  it  to  the  finding  of  areas 
and  volumes  are  given  before  the  formal  proof : 

To  find  the  area  of  the  figure  included  between  two  given 
ordinates,  the  axis  of   abscissas,  and  the  parabolic 
curve  whose  equation  is   a?=pyi 
Let  the  two  given  ordinates  corre-     y 
spond  to  the  abscissas  0Q,=a, 
and  OR,  =  h  ;  divide  qr  into  n 
parts  ;  let  the  abscissas  of  the 
n  -\-l  successive  points  (in- 
cluding Q  and  r)  be  Xq^x^^X2^ 
•  ••a;„;  and  the  corresponding 
ordinates  ^0. 2/1, 2/2,  •••  2/n»  and 
let  n  rectangles  be  formed  as  in  the  figure  ; 
then    •••  s,  the  area  sought,  is  the  limit  of  the  sum  of  5i,  Sg? 
^3?  **!  ^n,  the  areas  of  the  n  rectangles,  when  n  =  cx> 
and  Xi  —  Xq,X2  —  Xi,-"  each  =  0,  [th.  12,  nt. 

and     •••  Si  =  2/o-(»i— iCo)  [geom. 

=^'^^<^i-^o),  [hyp. 

,and    •.•  3xq^,  ^BxqXq^, 


x^^  —  x^ 


=  lim  — when  Xi—  Xq  =  0,       [§  5,  df .  deriv. 

•*a  —  •*'0 


19.  §12.]  INTEGRATION.  185 

wherein    q  is  some  variable  that  =  0  when  ajj—  iCo  =  0, 

and  Si       =  —  Ix,^—Xq^+(Xi-Xo)€i]. 

op 

So  ^2       =  —  [ojg^  —  Xi^  +  (X2  —  aJi)  €2] , 

«3  =  ^  W-  ^2^+  (a^S-  a'2)  €3]  , 

and  s„       =  —  [a;„»-  xi.i-\-  («„-  a;„_i)€„], 

3p 

wherein   ci,  eg,  •••  each  =  0  when  n  =  00  and  ajj—  a^o?  *••  ==  0  ; 

But     •.•  2(a?i— a?o)ci^(a?n— a;o)€^, 
wherein   c^  is  the  largest  of  the  €*s, 
.*.  2(a;i— aJo)ei==0  when  2s  =  s, 

=  —  (6^— a^).  Q.E.F. 

So  to  find  the  volume  of  the  solid  generated  by  the  same  figure 
revolving  about  ox,  the  tangent  at  the  vertex : 
then   •.*  V,  the  volume  sought,  is  the  limit  of  the  sum  of  v^^  Vg, 
%^  "•  "^ni  the  volumes  of  the  n  cylinders  of  revolu- 
tion generated  by  the  n  rectangles  when  n  =  cc  and 
a^i  — a^o, a^2— a?!,  •••each  =  0,  [th.  12 

and    '.•  Vi       =  Trt/o^aa— aJo)  [geom. 

=  5^'^^o'(^i-a;o)  [hyp. 

=  57»(S  +  '')^^-^>  [§5,df.deriv. 


186  INCO]SCMENSURABLES,   LIMITS,   ETC.  [VII.  th. 

and     •••  V.  =-!L-[(a;/-a'/)+(a;o-a;i)€2], 

op- 

op 

and  V    —-^^{y^—a^),  q.e.f.     [as  above 

5p' 

PKOOP   OF   THE    THEOREM. 

For     •. •  — — ^      =f'xo,  when  iCi—  iCo  =  0,         [§  5,  df .  deriv. 
•*'i —  **i) 

wherein   cj  is  some  variable  that  =  0  when  x^  —  Xq^^  0, 
•••A-/^o      =(a^i-aro)-(/a\,  +  €i). 

So  fX2-fXi  =(Xo-Xi)'(fXi-\-€2), 

fXs-fX2  ={Xs-X2)'{f'X2+€s), 

...  ...^ 

and  /a;„  -/a;„_i  =  («„-  a;„.i)  (/'a;„_i  +  €„)  ; 

wherein   cj,  cg,  •••  each  =  0,  when  w  =  oo  and  a^i—  Xq,  •••  =0, 

•••  f^n-fxo      ={Xi-Xo)fxo-\ l-K-a;n-i)/a;«-i 

H-(jCi-  a'o)ciH h  K-  aJn-i)en.     [li-  ax.  2 

But       •.•    '$(Xi-'Xo)€i^(x^—Xo)€^, 

wherein    e^  is  the  largest  of  the  c's, 

and     *.•  e^  =  0,  when  n  =  oc  and  a^i  —  rt'o,  •••  each  =  0,       [above 
.♦.  fb-fa=l\ml(xi-Xo)fxo+'"+{x„-x^_-i)f'x^_{\.Q.^.D. 

Note.   The  theorem  may  be  written  in  the  form  : 
\\m'tlf'x'mGX=fb—fa,  wheninca;  =  0, 

wherein   ^If'x- inc x=f'xQ'{ncxQ-\ f-/' ^n-i •  inc a!„_i, 

and  Xo  =  a,     aj„  =  &,     inca;^  =  a;^+i— a;^. 


19.  §  12.]  INTEGRATION.  187 

EXAMPLES   OF  THE    DIRECT   APPLICATION   OF   THE    THEOREM. 

To  find  V,  the  volume  of  a  segment  of  a  sphere  of  radius  r, 
whose    bounding    planes    are 
distant  a,  6  from  the  centre. 
Let  CDY  be  a  semicircle  of  radius  r ; 
take  CD  for  the  axis  of  x ;  let 
AF,  BE  be  two  ordinates  distant 
a,  h  from  the  centre  o  ;  and  let 
the  whole  revolve  about  cd  ; 
then  the  area  of  abef  is  the  limit  of  the  sum  of  the  areas  of  a 

large  number  of  rectangles  ; 
and  V,  the  volume  of  the  solid  generated  by  abef,  is  the 

limit  of  the  sum  of  the  volumes  of  the  corresponding 
cylinders  of  revolution. 

Take  Tryp^{Xp+i—Xp),  ='^(^— V)  (^p+i— ^p)?  ^s  the  type- 
term  of  this  series ; 
then    '.'  fx=Tr{r^—x^)^ 

.'.  fx   =7r{r^x  —  ia^), 
i.e.,         V     =^[(,^b-ib')-ir'a-ia')^  [v=/6-/a 

=  7r{b-a)[r'-i{a^+ab-\-b')2 

=  i-7rc(n2+r224-4r/), 
wherein  c  is  the  thickness  of  the  segment,  rj,  rg,  r^  the  radii  of 

its  bases  and  middle  section.  q.e.f. 

So  the  volume  of  the  hemisphere  generated  by  the  quadrant  ody 

=  7r(/r-/0)=7r(r«-ir^)  =f  Trr^ 
And  the  whole  volume  of  the  sphere 

=  |-7r?'^-  Q.E.F. 

Note.  If  in  ^7rc(?*i^-f-r2^-h4r3^),  the  general  expression  for 
the  volume  of  a  spherical  segment,  r  be  put  for  ri,  0  for  7\, 
•J-rVS  for  rg,  r  for  c,  the  result  is  the  volume  of  the  hemisphere  ; 
and  if  0  be  put  for  ri,  0  for  rg,  r  for  rg,  2  r  for  c,  the  result  is  the 
volume  of  the  sphere.  The  results  thus  found  are  identical  with 
those  given  above. 


188  IXCOMMENSUEABLES,   LOUTS,   ETC.  [VIL 

So  to  find  II,  the  height  fallen  through  in  a  given  time  by  a 
body  starting  from  rest,  on  the  assumption  that,  within 
any  distance  required  in  practice,  the  velocity  of  a  fall- 
ing body  increases  uniformly,  and  hence  that  the 
velocity  acquired  at  any  instant  is  proportional  to  the 
time  of  falling  from  rest ;  [laws  of  motion 

then         V  —  gt,  wherein  g  is  some  constant ;  v  is  the  velocity  at 
the  end  of  t  seconds  from  starting,  i.e.,  the  number 
of  feet  the  body  would  fall  through  in  the  next 
second  if  its  rate  did  not  change  during  that  second. 
Let  the  entire  time,  t,  be  divided  into  n  intervals,  ending 
respectively  at  <i,  ^2?"*^n  seconds  from  starting,  and 
let  ^0  =  0,  and  ?„  =  t  ;  assume  that  the  velocity  during 
each  intenal  of  time  remains  constant  at  what,  under 
the  laws  of  motion,  it  should  be  at  the  beginning  of 
the  interval,  and  let  ^i,  7^2,  •••  ^n  stand  for  the  distances 
fallen  through  in  the  1st,  2d,  •••  nth  intervals, 
then   •••  hi  =  Vo(ti—to)    =gto(ti—to), 
h  =  Vi(t2-  h)    =  gt^it.-  h) , 

.*.  H  =  g'-lim2o^-inc^         wheninci=0  [th.  19,nt. 

=  g.i{T^-0')  [f't==t,ft  =  if 

=  ^gT^.  Q.E.F. 

So  to  find  the  ultimate  average  (i.e.  the  limit  of  the  quotient 
of  the  sum  of  a  series  of  terms  by  their  number  when 
that  number  becomes  infinite)  of  the  successive  values 
taken  hyfx  as  x  increases  from  the  value  a  to  the  value  b  : 
Let  X  take  n  successive  values  a,  a-\-h,  a-j-2h,  •••6,  i.e.,  let 
a;  increase  by  n— 1  equal  increments  h  from  a  to  6, 
and  let  fx  take  the  corresponding  values  fa,  f{a  +  h) , 
/(a +  2 70,  '"fh, 

then    •.•  the  average  of  these  values  is  {fa  -\ f-/&)  :  ^, 

i.e.,  Qi'fa-\ Vh-fh)  '.{h  —  a-\-h),  \n—l=={h  —  a)  :  7i 

.♦.  theirultimateaverage,when7i=0,islim2^^'/flj:(6— a). 

[th.l9 


2. 


§  13.]  EXAMPLES.  189 

§13.     EXAMPLES. 

§  9.      PROB.  1. 

•  ••2.    Apply  the  increment  h  to  x  in  the  following  functions  ; 

find  the  corresponding  increments  of  the  functions, 
and  thence  their  derivatives  : 
1.   0?]  x"^'^  x-^\  x-^\  ax  +  h\  ax^—ho?\  ax-\-hx-^\  ax-^-\-hx^, 

—  5    —  >    — o  »    o —  >    5    7) j    -. — r-5 — • 

a    x    XT        Of  X  XT  {x  —  iy 

THEOR.  13. 

•  ••5.   Expand,  where  necessary,  and  find  the  derivatives  of: 

3.  2ar+3a;4-5;    (jk  +  I)  (a; +  2)  ;    o?+^ax^+^orx-\-a\ 

4.  x-\-x'^\    {x  +  x-^  +  iy-,    {l-x-\-x'Y-{l  +  x-x')\ 

5.  A+^Y;     «+A+    «    ;     A_l_    tY_A  +  l_J, 

\a     hj      X      Of      a?      \       X      mrj      \       x      xr 

6.  Of  what  are  the  following  expressions  derivatives  as  to  a;  ? 

Za?\    a;+2ar^+3a^;    ax  +  h\    x~^  \    ax'—bx^;   ax~^—bx~^, 

THEOR.  14. 

..•9.   Find  the  derivatives  of: 

7.  a;(a;+l);    -dx{ax-^b);    (2a;4-l)  (3aJ  +  2) ;   (3a;  +  l)2. 

8.  fl;(a;+l)(a;+2);    (a;  +  l)2(a;  + 2) ;    x\x+l);   x\x'^d). 

9.  x{a  —  2x){2a  +  Sx)',    (x  +  ay (x  +  by ;    (x^ay. 

10.    Of  what  are  the  following  expressions  derivatives  as  to  a;  ? 
2a;(a;+l)4-ic^;    3(2a;+l) +  2(3a;  +  2) ;    3(0.-2+3)  + 6 a;^ ; 
(a;+a)(a;  +  6)  +  (a;  +  6)(a;  +  c)  +  (a;  +  c)(a;  +  a). 

THEOR.  15. 

•  ••  12.   Find  the  derivatives  of: 

-^     1.  _«_,  a;+l,      X     ,  a;  — 1.  x-\-a ^   a  — a?,   a;^— aa;+5 
x^  x^  ^      X     '  aj+l'  a;+l'  x-\-b\a-\-x         x  —  a 

12     ^^-^.   ^-px-hq.        x-2  (x-j-2y^ix-iy 

a^     '  x^-^px  +  q'  x'^^x  +  e'   lx-2y^{x+iy 

13.    Of  what  are  the  following  expressions  derivatives  as  to  x? 
—  1  .   a;  — (a;+l).   b  (c -h  dx)  —  d  (a -\- bx)  ,     be  — ad 
ar^  '  x'         '  {c  +  dxy  '    {c  +  dx/ 


190  INCOMMENSURABLES,   LBIITS,  ETC.  [VII. 

THEORS.  16,  17. 

..•  17.    Find  the  derivatives  of : 

15.    (l  +  ar^)-;   {a^-x-^)'i;  ^{ax'+2bx-^c) ;  f{x^+px-}-q). 

•  ••  20.   By  finding  the  derivative,  d^,  of  the  first  derivative  d\ 
then  the  derivative,  d^,  of  d^,  and  so  on,  show  that : 

18.  D/(a^)  =  6;   D,^(iB*)  =  24;   D,''(a;")  =  w ! ;   Dj(af)  =  3Q0x'. 

19.  D,"»(x'')  =  ri(n— l)(n— 2)...(n-m+l)aJ—'*.    Find  d/ (»-«). 

2    //^  .     "N  «^  nl-a;      2(-l)".n! 

20.  D,V(^  +  ^0= i;    D«  7T~=  /I  ,    \n+i  ' 

{a^-{-a')^  l  +  a;       (l  +  xy^' 

21.  Of  what  are  the  following  expressions  derivatives  as  to  a;? 
4:{a-}-ba^y'3bj^;    12 6ar (a +  6a7^) 3;    -10a;(l-a^)^. 

22.  Show  that  x^-d^x  is  the  derivative  of  ;  thence  find 

n-\-l 

the  expressions  whose  derivatives,  as  to  x,  are  : 
2x{a^-j-x^y;   x^{a^+x^);    (ax  +  b)  ^{ax^+2bx-hc); 
Dj,x.           or— 2a;  +  2                   x  x^ 

X"'    (a^-3ar^+6a;-iy'    {a^ ^ ^)^'    {x' ^ x')^'' 
ax-^-b  .    a'3? -\-2bx-\-c 

§  10.     THEOK.  18. 

23.  By  means  of  h.  c.  msrs.  find  the  true  value,  when  a;  =  2,  of  : 
(a;2_5a.^(5)  .  (aj2_6a;  +  8);    {y?-%x-\-2)  :  (a^-4). 

24.  So,  when  a;  =  1 ,  of  : 

(a;3_3^.2_33._|_3)  .  (^^_^^^_rc_iy^    (x -1)  :  (x^-1). 

25.  So,  when  x  =  l,  of: 

(4^2-1)  :  (32af-l);    {6x'+a^-x)  :  (40:^- 6a^-4a;+3). 


§  13.]  EXAMPLES.  191 

2G.    So,  when  a;  =  c,  of  : 

(ax--2acx-{-ac^)  :  {bx^-2bcx+br)  ;    (a^-(^)^  :  (x^-c^) . 

27.  By  means  of  derivatives  find  the  true  value,  when  a;  =1,  of ; 
^of^^x'-2)  :(a^-f  2a^-2a;-l); 

(^x'-Saf+2a^+x-l)  :  {x'-x^-2x-{-2). 

28.  So,  when  a;  =  —  1,  of  : 

(ar^+1)  :  {x'-\-xr-\-x-\-l)  ;    (a^+1)  :  (a;5+4a;  +  5). 

29.  So,  when  a?  =  ^,  of  : 
(3a^-13a^4-23a;-21)  :  (6a;«+a^- 44a;  + 21). 

30.  So,  when  a;  =  2 a,  of: 

(.^4_  ^^'i_  ^2^2_  ^33.  _  2a^)  :  (3a;3_  7ax^-{-Sa^x  -2a^). 

31.  So,  when  a;  =  0,  of  : 

[i-V(i-^0] :  [V(H-^)-V(i  +  ^)]- 

32.  So,  when  a;  =  1,  of  : 

[(3a;3_2a;cyl_^!-|  .  [-i_a;f]  ;  [x^^ij^(x-l)^y[x'-l'\^. 

33.  So,  when  a;  =  a,  of : 

[(a2_a^)^  +  (a-a;)*]  :  [(a3-a^)^  +  (a -a^)^]  ; 

[V(«4-  a;)-V(2a;)]  :  [V(a  +  3a^)- 2  V^^]  ; 

[x-  —a-  +  (a;  —  a)  -]  :  [a^—  a-j  ;  [/a;— /a]  :  [<^a;— </>«]  ; 

34.  Put  a  +  7i  for  »,  a  +  7c  for?/,  expand  and  reduce,  then  let 

7i,  A;  become  infinitesimal,  and  thereby  find  the  true 
value,  when  x  =  y  =  a,  of  : 
[(a;-2/)a"+  {y-a)x''-\-  {a-x)y^}  :  [(x-y)  (y-a)  (a-x)']. 

§11. 

35.  Draw  the  lines  whose  equations  are  : 

a;  =  3;   y  =  6;   x=0;   y—x\   y—2x\   y  —  2x-\-'^. 

36.  Plat  the  equations  : 

(a;  +  3)2  +  2/2=:l6;   a^  +  92/'  =  9;   4aj2-/=16. 

37.  Trace  the  curve  whose  equation  is  a;?/  =  16  ;  and  show  that 

it  has  four  infinite  branches  that  continually  approach 
the  axes. 


192  IXCOIVESIENSUKABLES,   LIMITS,   ETC.  [VII. 

38.  In  the  equation  y=a^—6x^+2x-\-Q  give  x  the  values  —3, 

—  2,  —1,  0,  1,  2,  3,  4,  5,  in  succession;  thence  find 
the  corresponding  values  of  y,  and  plat  the  equation. 

39.  Represent  graphicall}^  the  functions  : 

3a;-f5;   ar^+l;   a:2  +  3a;  +  2;   x'-{-Sx  +  2i;  a.'2+3a;+3. 

40.  Plat  the  functions:  5  ±[9-(x+ 2)2]^;  5 ±[9  + (a? +2)2]^. 

41.  If  Po,(a;o^yo)»   i*i»(^»yi)    be  points  on  the  curve  2/ =/(a;), 

show  by  the  properties  of  similar  triangles  that  the 

equation  of  the  chord  PqPi  is 

(y-yo)'  (a;-aJo)  =  (!/i-yo)  :  i^i-^o)- 

Let  Pi  approach  Pq  so  that  a^  =  Xq^  t/i  =  2/o?  and  show  that 
the  equation  of  the  line  tangent  to  the  curve  at  Pq  is 
y  —  yo  =/'(^)  •  (a?  —  a\))  >  ^'9">  that  the  equation  of  the 
tangent,  at  the  point  whose  abscissa  is  2,  to  the  curve 
y  =  2a?  — 3a^  +  3a;  — 7    is   y  —  3  =  15(a;  -  2). 

Plat  this  curve  and  the  tangent. 

42.  Show  that  when  f'xQ  =  0  the  tangent  at  a^o,  yo  is  parallel  to 

the  a;-axis,  i.e.,  that  the  point  Xq,  yo  is  an  elboio  of  the 
curve  ;  and  that  when  the  plat  off'x  crosses  the  a>axis 
the  plat  of  fx  has  an  elbow. 

43.  Plat  the  function  a;''  — 4a^4-a^4-7a;  — 3  and  its  three  de- 

rived functions. 

§  12.     THEOR.  19. 

44.  Find    the    area   of    the   figure   bounded   by    the  axis  of 

abscissas,  the  curve  y  =  a^-\-x-{-l^  and  the  ordinates 
corresponding  to  the  abscissas  2  and  3  ;  find  also  the 
volume  of  the  solid  generated  by  the  revolution  of  this 
figure  about  the  axis  of  x. 

45.  So  for  ?/  =  (a;H-l)(a;  +  2),  between  the  abscissas  1,  3. 

46.  So  for  y  =  x^-\-4:a^-\-2aP-{-S,  between  the  abscissas  1,  2. 

47.  So  for  x^-\-  ax^-j-  a-x^-\-  b^y  =  0,  between  the  abscissas  a,  0. 

48.  Find  the  area  enclosed  by  the  axis  of  y^  the  lines  y=l, 

2/  =  0,  and  the  curve  x"^  {y^  -\-  5?/  +  4)  =  (2?/  -+-  5)^. 


§  13.]  EXA^IPLES.  193 

49.  Find  the  area  of  the  figure  enclosed  by  the  two  axes  and  the 

111 
curve  X'  -\-y^  z=a^ :  and  find  the  volume  of  the  solid 

generated  by  its  revolution  about  either  axis. 

50.  Find    the    area    of    the    figure    enclosed    by    the    curve 

^  5  3 

xy^  =  y'^  +  2 y'^  +  6  and  the  lines  x  =  0,  y  =  0,  y—1. 

51.  Find    the    area    of    the  figure   cut    oflf   from    the   curve 

y  =  (x-\-l){x-\-2)  by  the  axis  of  x. 

52.  If  the  figure  enclosed  by  the  curve  x^+y^  =  a^'  and  the 

axes  revolve  about  either  axis,  find  the  volume  of  the 
solid  generated. 

53.  Find  the  area  of  the  figure  cut  from  the  curve  ay^  =  a^  by 

the  line  x  =  a;  and  find  the  volumes  of  the  solids 
generated  by  its  revolution  about  that  line  ;  about  the 
axis  of  X ;  and  about  the  axis  of  y. 

54.  Find  the   volume    of   the   ring   generated    by  the   circle 

a^-{-y^=26  revolving  about  the  line  x=7. 

55.  The  curve  xy  =  (?  revolves  about  the  axis  of  y.     Show  that 

the  volume  generated  by  the  infinite  branch  beginning 
at  the  vertex  (c,  c)  is  equal  to  the  volume  of  the 
cj'linder  generated  by  the  ordinate  of  the  vertex. 

56.  Find  the  ultimate  average  value  of  the  function  3ic^+5a?— 7 

as  X  varies  continuously  from  1  to  4. 

57.  So  for  the  function  x^—  3 o:^  4-  2  £c  —  1  between  a;  =  0  and  3. 

58.  If  the  function  named  in  Ex.  57  be  platted,  show  that  the 

result  of  that  example  gives  the  ultimate  average 
length  of  equidistant  ordinates  between  a;  =  0  and 
x=?>. 

59.  Find  the  average  ordinate  lying  between  the  given  ordinates 

in  Examples  44-47 ;  and  show  that  for  any  figure 
with  a  rectilinear  base  the  product  of  the  average 
ordinate  by  the  base  is  the  area. 


194  POWEKS   AND   ROOTS.  [VIII. 

Ylll.    POWERS  AND  ROOTS. 

§1.    FRACTIONAL  POWERS. 

The  words  j^oiver^  root,  base,  exponent,  and  root-index  are 
defined  in  I.  §  10.  A  root-index  is  always  assumed  to  be  a 
positive  integer ;  but  an  exponent  may  be  any  number  whatever. 

The  value  of  a  fractional  power  is  commonl}'  ambiguous. 

E.g.,    100^  =  ±10;     9"^  =  ±3^. 

So,  as  appears  later,  ever}-  base  except  0  has  three  distinct 
cube  roots,  four  distinct  fourth  roots,  and  so  on.  Some  of 
these  roots,  however,  are  neither  purely  positive  nor  purely 
negative  ;  they  are  called  imaginaries,  or,  better,  complexes,  and 
discussed  in  chap.  X. 

Different  powers  of  a  base  are  in  the  same  series  when  they 
are  integral  powers  of  the  same  root.  An  integral  power  of  a 
base  belongs  to  all  series  alike. 

E.g.,   d-\    9-^   9«,    9^   9\   d\  9^    ... 
are  the— 2d,  —1st,    0th,  1st,  2d,  3d,  4th, ...  powers  of    V^, 
i.e.,  of  ~3  and  of  '^3  :  they  form  the  two  series 

i,      -|,       1,    -3,     9,  -27,  81,    ...,  powers  of  "V^, 
and  i,        J,       1,     3,     9,     27,81,   •..,  powers  of  ^V^ ; 

but  the  integral  powers  ^,  1,  9,  81,  belong  to  both  series. 

When  several  powers  of  the  same  base  occur  together,  they 
are  assumed  to  be  all  taken  in  the  same  series. 

E.g. ,   the  value  of   4~  ^  —  3  •  4^  H-  4^   is  either 

^_3.+2+(+2)3,  =   21,  or-L-3.-2+(-2)^  =-2i 

according  as  4"^,  4-,  4^  are  taken  as  powers  of  "•'2  or  of  ~2  ; 
butnot    :^-3.-2+(+2)3,  =:14^,nor-^-3.+2  +  (+2)3,  =     1}. 

So,       -yyda^--^'4:a^=±Sa^:2a  =  ±a, 
butnot    3a-\-2a,     nor    —3  a  — 2  a. 

Powers  of  different  bases  are  like  powers  if  they  have  the  same 
exponent. 

^'9;   V«'   V^'  V«^;    «^  ^^  «^^    2",  3",  6". 


§  2.]         COMBINATIONS  OF  CO]NLMENSUEABLE  POWERS.        195 

§2.    COMBINATIONS  OF   COMMENSURABLE  POWERS. 

1^^  That  every  commensurable  power  of  a  real  positive  base 
has  at  least  one  real  value  is  shown  from  independent  consider- 
ations in  th.  5,  which  may  therefore  be  read  here  if  preferred. 

Theor.  1.  Any  commensurable  poicer  of  a  base  Jias  the  same 
value  or  values  whether  the  exponent  be  in  its  lowest  terms  or  not. 

Let  A;,  p\  q  be  any  positive  integers  ;  j9,  =  ±p',  any  integer, 
positive  or  negative  ;  a,  any  base  ; 

kp  p 

then  will  every  value  of  a**  be  a  value  of  a*,  and  conversely. 

For     •••  *^A  =  one  of  any  kq  equal  factors  into  which  a  can  be 
resolved,  [I.  §10df.root 

and     *.•  the  product  of  any  k  of  these  factors  is  equal  to  that 
of  any  other  k  of  them, 
.'.  all  the  q  partial  products  so  formed  are  equal,  and 
each  is  a  value  of  -^a, 

and  every  single  value  of  *^a  is  a  value  of  -^(-^a). 

So  •••  l^{V^)T=L</(-V^)l  -[a/CVa)]  •••A:g  factors 
=  [^(V^)]*-[^(Va)]*-  g  factors 
=  ( Va)  •  ( V^)  •  (-y^)        •••   g  factors 

=  A, 

.*.  every  value  of  -s/{-^a)  is  a  value  of  ^a.         [df.  root 

i.e.,  every  value  of  either  member  is  a  value  of  the  other. 

So      •.•  a*'  =  [^(-^a)]^p  [I.  §10df.  fract.  pwr. 

=  ^^L</(-V^)l  "^[^(Va)]  >f-"kp' times 
=  ^^LV(^^)Y'^l</(-\/^)Y^"'  P' times 

=  1^(-</a)  ^(Va)  ^••.  i^'times 

p'  p 

=  A   *  =A% 

.'.  every  value  ofeither  member  is  avalue  of  the  other,  q.e.d. 

Note  1.    In  general,  when  either  member  of   an  equation 

admits  more  than  one  value,  the  sign  of  equahty  asserts  that 

every  value  of  either  member  is  a  value  of  the  other. 

p 
Note  2.   In  what  follows  +A*  =  the  positive  value  of  -^a^. 


196  POWERS   AND   ROOTS.  [VIII. 

PRODUCT    OF    LIKE    TOWERS. 

Theor.  2.  Every  value  of  the  product  of  like  commensiirable 
powers  of  two  or  more  bases  is  a  value  of  the  like  power  of  their 
product. 

Let  A,  B,  c,  •••  be  any  bases,  and  a",  b",  c",  •••  like  commen- 
surable powers  of  them ; 
then  will  every  value  of  a^-b^'C"  •••  be  a  value  of  a -b  •€•••". 

(a)  n  an  integer^  positive  or  negative.  [II.  th.  3  cr.l2 

(b)  n  a  simple  fraction. 

For  let  71=^,  wherein  </  is  a  positive  integer,  p  an  integer 

either  positive  or  negative  ; 
pi  pi  pi 

then    •••  a»=(a«)'',    b«  =  (b9)p,    c?  =  (c5)^,  •••      [df .  f ract. pwr. 
p     p     p  111 

.-.  A»-B«-c^---     =  (x^y •  {B^y •  (c^y •  " 
111 

=  (a«.b^-c«---)^.  [(a) 

But    •.•   (A«.B«.c^-..)'=(A«)'-(Be)«.(c^")*...  [(a) 

=  A'B'C---, 

111  1 

.-.  every  value  of  a«'B^-  €«•••  is  a  value  of  (a-b  •  c---)? ; 

[df . root 
p     p    p  111 

.*.  every  value  of  A?  •  B» •  c?  ••• ,    =  (a«-b?«c*'«-)^,     [above 

p 
is  a  value  of  (a 'B •€•••)«.  q.e.d. 

Note  1.    In  the  demonstration  of  case  (6)  nothing  need  be 

p     p 

said  of  the  series  to  which  the  powers  a*?,  b?,  •••  belong ;  for  the 
demonstration  holds,  and  the  theorem  is  true,  whichever  values 
of  the  roots  of  a,  b,  •••  be  taken. 

E.g.,   of  42-,  9^  36^  the  values  are  "8,  -27,  "216, 
whereof  the  product +8  • +27,    or    -8- "27,  is   +216, 
and  the  product -8. +27,    or   +8- "27,  is    -216. 

Note  2.  When  the  exponent  is  fractional  and  some  of  the 
bases  are  alike  or  so  related  that  their  powers  must  be  in  the 
same  series,  the  product  of  the  powers  may  admit  fewer  values 
than  the  power  of  the  product.  [comp.  X.  th.  7  nt. 

E.g.,    V«- V«- V^'V4^^=+2a&2only, 
but  V^a^ft^  =±2a62. 


§  2.]        COMBINATIONS  OF  COMMENSURABLE  POWERS.        197 

Cor.  Every  value  of  the  quotient  of  like  commensurable  pow- 
ers of  two  bases  is  a  value  of  the  like  power  of  their  quotient. 

E.g.,    ^a:^'b  =  ±^{a:b)',     ^a:  ^4.a  =  +  l  :^4  =  %. 

PRODUCT   OF   POWERS   OF   SAME    BASE. 

Theor.  3.  The  product  of  two  or  more  commensurable  poicers 
of  any  same  base,  in  any  same  series,  is  that  power  of  the  base 
whose  exponent  is  the  sum  of  the  exponents  of  the  factors. 

Let  A"*,  A**,  •••  be  any  commensurable  powers  of  a  base  a,  in 
the  same  series ; 
then  will  a'^-a"---  =  a'"  +  "  +  -. 

(a)  m,  n,  •••,  all  integers,  icJiether  positive  or  negative. 

[II.  th.3cr.10 

(b)  m,  n,  •••,  some  or  all  of  them  simple  fractions. 

For  let  m  =^,  ?i  =  -,  •••,   wherein  »,g,  r,s,  •••  are  all  integers, 
q  s 

and  the  denominators  g,  s,  •••  are  all  positive  ; 

Y)      ct    r      h 
and  let  k  be  the  I.e.  mlt.  of  g,  s,  •••,  so  that  -  =  -,  -  =  -,•••; 

q      k    s     k 

then   •.•  a?=a*  =  (a*)^,    a»  =  a*  =  (a*)*..-,  [th.l,df.fract.pwr. 

1 
wherein   a*  preserves  throughout  the  same  value,     [same  series 

p       r  11 

.-.    A?.A«---    =(a*)^-(a*)*... 

=  (Jcy+^+"'  [(a) 

=  A     *  [df .  fract.  pwr. 

=  A*     *         , 
i.e.,  a'^-A"-.  =  A"'  +  "  +  "*.  Q.E.D. 

Cor.  1 .  TJie  quotient  of  tivo  commensurable  powers  of  any  same 
base,  in  any  same  series,  is  that  power  of  the  base  whose  exponent  is 
the  excess  of  the  exponent  of  the  dividend  over  that  of  the  divisor. 

Cor.  2.  Of  two  or  more  commensurable  powers  of  any  same 
base,  in  any  same  series,  the  product  or  quotient  is  in  the  same 
series. 


198 


POWERS  AND   ROOTS. 


[VIII. 


POWER   OF   A    POWER. 

Theor.  4.  A  commensurable  power  of  a  commensurable  power 
of  any  base  equals,  or  includes  among  its  values,  that  power  of 
the  base  whose  exponent  is  the  product  of  the  given  exponents. 

Let  A  be  any  base,  and  m,  n  any  commensurable  numbers  ; 
then  will  (a"*)"  have  every  value  of  a""*. 

(a)  n  a  positive  integer. 
For  (a'")'*  =  1- a'"-a'*---,  71  times 


.  (m  +  m+ •••,  n  times) 

a"*** 

(6)  n  a  negative  integer. 
For     •••   —  n  is  then  a  positive  integer, 

.*.   (a'")'*=  1  :  a"*:  A"*:  • n  times 

=  1 :  (a™ "A*" n  times) 

=  1 :  A'"(-"> 
=  1  :  A"""* 


[I.  §  10  df .  int.  pwi'. 
[th.3 

Q.E.D. 


[df.  int.  pwr. 

[II.th.3cr.8 

[th.3 


Q.E.D.     [df .  commens.  pwr. 


(c)  n  a  fraction  -  ;  p,  q  integers,  q  positive, 

tn  A 


For 


(Af)' 


[(«) 

[df .  root 
[(«,6) 


.-.  every  value  of  a*  is  a  value  of  -^(a"*), 

p  m  m 

.'.  every  value  of  A*" ■  ?,    =a2"^,    =(a«)p, 

is  a  value  of  [VC^*")?* 
p 
i.e. ,  it  is  a  value  of  (a"*)  *,  [df .  f ract.  pwr. 

.-.  every  value  of  a'"'*  is  a  value  of  (a'")''.  q.e.d. 

Cor.  1.  If  m,  nbe  commensurable  numbers,  (a'")°  and  (a°)™ 
have  at  least  one  value  in  common,  viz.,  a"°. 

CoR.  2.  In  particular,  if  b  =  a~^,  then  b  p  has  the  value  a 
whether  or  not  it  have  other  values  also. 

Cou.  3.  The  reciprocal  of  any  poicer  of  a  base  is  the  like 
power  of  the  reciprocal  of  the  base; 

[II.th.3cr.8 


i.e.,  1  :A~«  =  l^A^>i;  A«>ic  •••i^  times, 

=  (1:a)-^"     or     (a-I')-i  = 


(.-^fi. 


§  3.]  CONTINUITY  OF   COMMENSURABLE  POWERS.  199 

Note.    If  n  be  a  fraction  ^,  and  q  be  not  prime  to  m,  or  to 

its  numerator  if  m  be  a  fraction,  then  (a"*)"  may  have  values 
not  included  among  those  of  a""*. 

E.g.,    (102)2  has  not  alone  the  value  103=+1000  ; 
but  also   (10^)^=  (100)  ^  =  (100^)3  =  (-10)3  =  -1000. 

So,        (^-yjxY^   x^     = +a;2  only  ; 
but  y/x^     =  (a.**)  ■'  =±x^.  [comp.  X.  th.  9  nt. 

§3.    CONTINUITY  OF  COMMENSURABLE  POWERS. 

Theor.  5.    If  there  he  a  positive  base  A-j   ^'^^J^J    than  unity, 
then : 

1.  For  every  positive  commensurable  exponent  n  the  power  a° 
has  one  positive  value  +a°<J   "'^f?     than  unity; 

2.  For  every  negative  commensurable  exponent  n  the  power  a° 
has  one  positive  value  +a°^  '^^^  f^  than  unity; 

3.  In  either  case,  a°  has  but  one  such  value, 
1.    n  positive. 

(a)    n  a  positive  integer. 
For     •.•  A"  =  1 -A'A*-,  ?i  times, 
and     • .  •  each  of  the  factors  a  is  positive  and  ^  ^  1  ? 

.-.  the  product  a"  is  positive  and^  ^1.  q.e.d.  [II.§3ax.l9 

(5)  n  the  reciprocal  of  a  x^ositive  integer  q. 

For  let  ic,  a  variable,  ■{   ,       '       continuously  from  the  value 

1  to  the  value  a  ; 
then   • .  •  inc  iB« :  inc  a;  =  deriv.  a;^,  =  qxf^-^,  as  inc  a;  =  0 ,  [ VII.  th.  1 7 
.'.  when  inca;  is  infinitesimal,  so  is  incaj';    [VII.th.2cr.l 
i.e.,  as  X  passes  continuously  through  all  values  from  1  to  a, 

xf^  takes  every  value  between  1  and  a^  ; 

but     •••  a',  =  1  •  A' A"-,  g  times, ^  ^AwhenA^  ^1, 

.*.  A  is  a  value  between  1  and  a^  ; 
.'.  x'^  passes  through  the  value  a, 

and  jJ  has  the  value  x,  a  positive  number^       1.  q.e.d. 


200  POWEKS  AND  ROOTS.  [VIII. 

(c)   n  any  positive  fraction  L. 

4 

For     •.•  A?  has  a  positive  value ^  ^1,  [(/>>) 

.-.  A«,  =  (a«)',  has  a  positive  value ^  ^1.     [q.e.d.   [(a) 

Note.     a«  is  not  necessarily  commensurable,  even  when  a  is 
commensurable. 

2.  n  negative. 

Let  n,  the  exponent,  =  —  m,  wherein  m  is  positive  ; 
then   •.•  A™  has  a  positive  value  "^a*",  ^  ^1^  [(1) 

.-.  A**,  =  1 :  A"*,  has  a  positive  value  "^a",  ^  ^  1-     Q-e.d. 

3.  But  one  positive  value. 

1 
For  if  possible  let  the  root  a»  have  two  positive  values  un- 
equally large ; 

then         the  product  A,   =1.a«-a«---  q' times,  has  two  values 
unequally  large,  [II.  §3,  ax. 19 

which  is  contrary  to  the  hypothesis  ; 
1 
.*.  A«  has  but  one  positive  value, 

.'.  the  product  a*',    =1.a*-a«---  p  times,  has  but  one 

positive  value. 

p  p 

So  with  a"«,    =  1  :  a?. 

Cor.    If  the- base  and  exponent  be  both  finite,  so  is  the  x>ositive 
value  of  the  power. 

(a)    The  exponent  an  integer,  either  positive  or  negative. 
For     •••  the  power  is  the  continuous^  ^      ,.     ,of  1  by  a  finite 

number  of  finite  {  ^"l^oP^f ''  [df  •  int.  pwr. 

.-.  the  power  is  finite.  q.e.d.      [VII.th.llcr.3 

(6)    The  exponent  a  fraction. 
Let  the  base  a  <(  ^  1 ,  and  let  the  exponent  n  lie  between  the 

integers  i  and  i  +  1  ; 
then    •.•  n  —  i  and  t  +  1  —  n  are  positive  and  commensurable, 

.-.  A"  S  A^+^-%  both^  ^1;  [th.5 


§  3.]  CONTINUITY  OF  COMMENSURABLE  POWERS.  201 

and     •••  a'^  =  a^^-'.aS    a*+i  =  a*  +  i-".  a%  [th.3 

.    .    A    ^    ^A,  A         ^    ^A  , 

i.e.,  '^A''  lies  between  a*  and  a*+^,  which  are  both  finite  and 

positive.  [(a) 

.•.  +A"  is  finite.  q.e.d. 

Theor.  6.  Of  a  commensurable  {-^  .  power  of  a  variable 
positive  base  ivith  given  exponent,the positive  value  is  ^  ,  ,  .  ^ 
continuous  function  of  the  base. 

Let  A  be  the  variable  base,  and  n,  =  ±  -,  the  given -j^  ^  J 
exponent ;  then : 

(a)    The  larger  the  base  a,  the'{       ^,,     the  power, 

1     1 
For     *.•  A  =  1- A?-A?-"  5' times, 
1 
.*.  the  larger  a?  is,  the  larger  is  a  ;  [II.  §  3  ax.  19 

i.e.,  the  larger  a  is,  the  larger  is  a^; 

p  1      1 

and    • .  •  A!?  =  1  •  A«  •  A^  •  •  •  I?  times, 

1  p 

.*.  the  larger  ai  is,  the  larger  is  a*  ; 

p 
.*.  the  larger  a  is,  the  larger  is  a^  ; 
p  p 

and  the  smaller  is  a"*,    =  1  :  +a'  ; 

,p  _p 

i.e.,  when  A  increases,  A  *  increases,  but  a  ^decreases,  q.e.d. 

(6)   Whe7i  A  passes  through  every  value  from  0  to  "'"go  in  order, 

"^A°  passes  through  every  value  from  \  j^      .      a    ^^  order. 

For,  let  B  be  any  positive  number ; 


then 


and 


the  power  B"  has  a  positive  value  +b»,  [th.  5 

1  1 

if  A  =  +B^,  +a",  =  +(+b«)",  takes  the  value  b,       [th.4 

every  number  b  from  0  to  "*'go  becomes  in  turn  a  value 

of  +A'' ; 
the  larger  the  base,  the  ^    arger    ^he  power,  [(a) 

when  A  passtis  through  every  value  from  0  to  +co  in 
order,  "^a"  passes  through  every  value  from  ■{  +  ,  q 
in  order,  i.e.,  it  is  continuous.  q.e.d. 


202  POWERS  AND  ROOTS.  [VIII. 

Cor.  1.  If  the  base  and  exponent  be  both  finite^  every  injini- 
tesimal  change  in  the  base  gives  an  infinitesimal  change  hi  the 
positive  value  of  the  power ^  and  conversely. 

Cor.  2.  If  the  base  approach  a  limit  Aq,  the  positive  value  of 
the  power  approaches  a  limit  +Ao°,  and  conversely. 

Theor.  7.  Of  a  variable  commensurable  power  of  a  constant 
positive  base  larger  than  unity : 

1 .  The  positive  value  is  an  increasing  f  Miction  of  the  exponent ; 

2.  Tlie  exponent  can  be  so  taken  that  the  power  shall  lie  be- 
tween any  two  positive  numbers^  however  close  together. 

Let  A,  >1,  be  the  base ;  and  let  n\  n"  be  any  two  values  of 
the  exponent  ?i,  such  that  n'<  n"  ;  then  : 

1.  "'"A"  is  a7i  increasing  function  of  n. 

For     •.•  +A''''  =  +A"'-+A""-'''  [th.3 

=  ■•"A"'  •  a  positive  power  of  a 

=  ^A"'  •  a  number  greater  than  1 ,  [th.  5  ( 1 ) 

.-.  +A''">+A"'; 
I.e.,  ^A"  increases  with  w.  q.e.d. 

2.  n  can  be  so  taken  that  +a"  shall  lie  between  any  positive 
numbers  b,  c,  whereof  b  <  c. 

1     c 
Take  a',  q  any  positive  integers  so  great  that  a'>a,  -<  —  1; 

1  Q      B 

and  let  h  =  — — ; 

a'q 

1    ^'**       *            1 
then   •.•   (1+-)      =1  +  a'q f- other  positive  terms   [bin.th. 

Q  Q 

>1+a' 

>A, 

.-.A*  <1  +  1  [th.  6(a) 

Q 

<■' 

.'.  of  the  series  •..+a-3%  +a-2%  +a-*,  +A^  +a\  -^a'\  +a^\ 
•  ••  each  term  is  less  than  the  -th  part  of  the  term 

B 

next  before  it. 


§  3.]  CONTINUITY  OF  C0:^I3IENSURABLE  POWERS.  203 

But  this  series  has  terms  >b  ; 

for,  if  mh  be  any  integer  > — "—^ 

then  A'"%  =(1H-A=T)'"%  >l+m7i(A-l)  >b.  [bin.  th. 

And  the  series  has  terms  <  b  ; 

for,  if  A"*'*  be  any  term  >-■ . 

B 

then  A"'"''^,   =  1  :  a"*'*,  is  a  term  <  b. 

Let  A^*  be  the  greatest  term  of  the  series  less  than  b  ; 

then         the  next  term,  a^^+^)*,  <--b  or  c,  [above 

B 

i.e.,  A^^+^^*,  a  commensurable  power  of  a,  lies  between  b 

and  c.  Q.E.D. 

CoR.  1.  Of  a  variable  commensurable  poiver  of  a  constant 
positive  base  smaller  than  unity ^  the  positive  value  is  a  de- 
creasing function  of  the  exponent,  and  can  be  made  to  lie  between 
any  two  positive  numbers. 

Let  the  base  be  a,  =1 :  a,    wherein  a  >1  ;    and  let  6,  =  1  :  b, 
and  c,  =  1 :  c,  be  any  two  positive  numbers  ; 
then   *.*  +A"  increases  with  the  exponent,  and  takes  values  be- 
tween the  positive  numbers  b,  c,  [th.  7 
and     •.•  +a"  is  the  reciprocal  of +a",                               [th.4cr.3 
.*.  +a"  decreases  as  the  exponent  increases,        [II.  ax.  18 
and          takes  values  between  b  and  c.                         q.e.d. 

Cor.  2.  When  the  base  differs  sensibly  from  0,  1,  and  oo,  and 
the  exponent  is  not  oo,  then  every  infinitesimal  change  in  the  ex- 
ponent gives  an  infinitesimal  charige  in  the  positive  value  of  the 
power,  and  conversely. 

CoR.  3.  Wlien  the  base  differs  sensibly  from  0,  1,  and  oo,  and 
the  exponent  approaches  a  limit  no,  the  positive  value  of  the 
power  approaches  a  limit  +a°o  ;  and  conversely. 


204  POWERS   AND   ROOTS.  [VIII. 

Note.  The  principles  established  in  theors.  5,  6,  7  are  sum- 
marized as  follows : 

Every  commensurable  power  of  a  positive  base  has  one  and 

,    .  ,         ...  7       rxi    KT      i  ot**  increasinq 

out  one  real  positive  value  [th.  5]  ;  -j      ^    ,      .    ^  continuous 

function  of  the  base  if  the  base  vary  and  the  exponent  be  constant 

if  the  exponent  vary  and  the  base  be  constant  and  -{         jj  ,  than 
unity  [th.  7,  th.  7  cr.l]. 

WJiether  the  base  or  the  exponent  varies,  the  commensurable 
power  takes  values  [indeed,  an  infinite  number  of  them]  between 
any  two  positive  numbers  liowever  close  together  [above,  th. 7,  cr.l] . 

When  both  base  and  exponent  are  finite,  and  the  base  ^  1,  any 
infinitesimal  change  in  either  gives  an  infinitesimal  change  in  the 
positive  value  of  the  power,  and  conversely  [th.  6  cr.l,  th.7  cr.  2]  ; 
and  if  either  the  base  or  the  exponent  approach  a  limit  while  the 
other  is  constant,  so  does  the  positive  value  of  the  power,  and 
conversely  [th.  6  cr.  2,  th.  7  cr.  3] . 

The  positive  value  of  the  power  is  finite  when  the  base  and  the 
exponent  are  finite  [th.5  cr.].   This  value  is  {      ^  J.  ,  than  unity  if 

the  exponent  be  positive,  and  ■{  .         .    than  unity  if  the  exponent 
be  negative,  when  the  base  is-{       ^,,     than  unity  [th.5]. 

It  appears  later  [th.  12]  that  the  powers  of  a  constant  base 
take  a  continuous  series  of  values  when  the  exponent  takes  a 
continuous  series.  But  when  the  varying  exponent  or  base  is 
restricted  to  commensurable  values,  then  between  any  two 
values  taken  by  the  power  there  lie  an  infinite  number  of  values 
not  so  taken. 


§4.]  INCOMMENSURABLE  POWEES.  205 

§4.    INCOMMENSURABLE    POWERS. 

Hitherto  no  meaning  has  been  given  to  the  symbol  a"  when 
n  is  incommensurable  ;  and  any  meaning  that  may  now  be  given 
to  it  should  fulfil,  if  possible,  the  following  conditions : 

1 .  It  should  give  a  single  definite  positive  value  to  the  symbol 
A"  when  A  has  a  given  positive  value  and  n  is  incommensurable. 

2.  It  should  not  conflict  with  any  use  that  the  symbol  a"  has 
when  n  is  commensurable. 

3.  It  should  preserve  all  the  fundamental  properties  that 
the  symbol  a**  has  when  n  is  commensurable :  in  particular, 
theorems  2,  3,  4  should  be  true  for  all  real  exponents  whatever. 

The  following  theorem  lays  a  foundation  for  the  definition : 
Theor.  8.    If  there  be  a  constant  positive  base  not  0  7ior  1 
nor  oc,  and  two  variable  commensurable  exponents,  one  increas- 
ing and  the  other  decreasing  toward  a  common  incommensurable 
limit  not  go,  then : 

1 .  The  positive  values  of  the  two  variable  commensurable  powers 
have  a  common  limit,  which  lies  between  them  and  is  not  0  nor  qo  . 

2.  This  common  limit  depends  upo7i  the  value  of  the  base,  and 
of  the  cow.mon  limit  of  the  exponents,  but  not  upon  the  law  by 
which  either  exponent  approaches  its  limit. 

3.  This  common  limit  is  not  a  commensurable  power  of  the  base. 
Let  A  be  any  constant  positive  base  not  0  nor  1  nor  qo  ;  let 

X,  y  be  any  variable  commensurable  exponents,  x  increasing  and 
y  decreasing  toward  a  common  incommensurable  limit  n  not 
infinite ;  and  let  x\  y'  stand  respectively  for  a  particular  series 
of  values  of  x,  y  that  approach  n  as  their  common  limit,  and  so 
with  x",  y'\  with  a;'",  ?/'",  •••,  then: 

1.  +A^',  ^A^'  have  a  common  limit  that  lies  between  them  and 
is  not  0  nor  co. 

For     •.*  the  exponents  x',  y'  each  =  n,  their  common  limit,  [hyp. 
.'.  x',  y'  come  to  differ  from  each  other  by  less  than  any 

assigned  number, 
.*.  +A*',+A^'  come  to  differ  from  each  other  by  less  than 
any  assigned  number  ;  [th.  7  cr.  2 


206  POWERS  AND   ROOTS.  [VIII. 

and     •.•  x'<y'  always,  and  x'  increases  while  y'  decreases, 

.-.  ^A"^  <-A>'   always,   and  ^a"  ^  ™;;^  while  ^a-' 

,  decreases,  ^         ,  >j  [th.7(l), th.Tcr.l 

'  increases,  '  <  •-        ^  ^' 

.'.  +A'',  "'"A'''  approach  a  common  limit  that  lies  between 
them,  and  therefore  is  not  0  nor  oo.  q.e.d.  [Vll.th.l 

2.  The  variable  powers  +a*',  ^a^',  +a''",  '^a^",  •••  have  the  same 
common  limit. 

For  the  variable  powers  "'"a''",  ^a"'  have  a  common  limit,  the 

same  as  the  limit  of  '•"a"', 
i.e.,  the  same  as  the  common  limit  of  '•'a''',  "^a"'. 

So        the  variable  powers  '•'a''",  ^a" "  have  a  common  limit,  the 

same  as  the  limit  of  +a*",  +a^'  ;  and  so  on. 

3.  This  common  limit  of  "'"a'',  +a^  is  not  a  commensurable 
power  of  A. 

For,  if  possible,  let  this  limit  be  some  commensurable  power,"*" a*"; 
then  is  the  commensurable  exponent  m  the  common  limit  of  the 
variable  commensurable  exponents  cc,  y,  [th.7  cr.3  cnv. 
which  is  contrary  to  the  hypothesis  ; 
.*.  this  supposition  fails,  and  it  is  only  left  that  the  com- 
mon limit  of  "'"A'',  "^a",  •••be  not  a  commensurable 

power  of  A.  Q.E.D. 

DEFINITION. 

If  there  be  two  variable  commensurable  powers  of  a  given 
base,  the  one  increasing  and  the  other  decreasing,  and  such  that 
their  variable  exponents  have  a  common  incommensurable  limit 
that  lies  between  them,  then  the  symbol  formed  by  writing  the 
base  with  this  limit  for  exponent  stands  for  the  common  limit  of 
the  positive  values  of  the  variable  powers  and  is  an  incommen' 
surable  j)ower  pf  the  base. 

That  this  definition  satisfies  the  first  two  of  the  conditions 
stated  above  is  evident  from  theor.  7  ;  and  that  it  satisfies  the 
third  condition  appears  from  the  theorems  that  follow. 

Note.  It  appears  later  that  a**  may  have  all  the  limiting 
values  of  a*,  a",  i.e.,  every  value  of  a*  =  some  value  of  a**  ;  but 
only  the  positive  limiting  values  are  considered  here. 


§  5.]  COMBINATIONS   OF  POWERS  IN   GENERAL.  207 

§5.    COMBINATIONS  OF  POWERS   IN  GENERAL. 

PRODUCT    OF    LIKE    POWERS. 

Theor.  9.     Every  value  of  the  product  of  like  powers  of  two 
or  more  bases  is  a  value  of  the  like  power  of  their  product. 

Let  n  be  any  number  and  a,  b,  c,  •••  be  any  bases  ; 
then  is  every  value  of  a'*  •  b"  •  c"  •  •  •  a  value  of  a-E'C'"". 

(a,  b)  n  commensurable.  [th.  2 

(c)        n  incommensurable. 

For  let  a;  be  a  commensurable  variable  whose  limit  is  n  ; 
then   •.•  A*  =  A'',    b*=b",    c'=  =  c",  •••,  [df.  incom.  pwr. 


and 


A^^.B^-C".. 

.  =  A".B'».c«...;                              [VILth.8 

A^'-B^-C^.. 

.  =  a  value  of  A-B-C"-"                          [th.  2 

=  A-B-c---",                          [df.  incom.  pwr. 

A*'-B''.C"-. 

•  =  a  valueof  A* B •€•••".  q.e.d.  [VII.  th. 6  cr. 

CoR.  Every  value  of  the  quotient  of  like  poioers  of  two  bases 
is  a  value  of  the  like  power  of  their  quotient. 

PRODUCT   OF   POWERS   OF    SAME    BASE. 

Theor.  10.  The  product  of  tico  or  more  powers  of  any  same 
base,  in  any  same  series,  is  that  power  of  the  base  whose  exponent 
is  the  sum  of  the  exponents  of  the  factors,  and  in  the  same  series. 

Let  m,  n,  •••  be  any  numbers,  and  a  any  base ; 
then  will  a"*- a'*.--  =  a'™+'*+-. 

(a,  6)  m,  n,  •••  all  commensurables.  [th.  3,cr.  2 

(c)        m,  n,  •••  so7ne  or  all  of  them  incommensurable. 

For  let  X,  ?/,•••  be  commensurable  variables  whose  limits  are 
m,n,  •••  respectively, 
then    • . •  A''  =  A"*,     A^  =  A",  •  •  • ,  [df.  incom.  pwr. 

.-.    A'=.A^...=A"*-A^..-.  [VII.  th.  8 

But     •••  A==.Ay...=A"'+y  +  -  =  A'"+"+-,      [th.  3,df.  incom.  pwr. 
.-.  A"*.A'»...  =  A'^+"+-.  Q.E.D.     [VILth.  6cr. 

CoR.  Tlie  quotient  of  two  powers  of  any  same  base  in  any 
same  series  is  that  power  of  the  base  whose  exponent  is  the  excess 
of  the  exponent  of  the  dividend  over  that  of  the  divisor. 


208  POWERS  AND   ROOTS.  '  [VIU. 

POWER    OF    A    POWER. 

Theor.  11.  A  power  of  a  power  of  any  base  equals^  or  in- 
cludes among  its  values,  that  power  of  the  base  whose  exponent  is 
the  product  of  the  given  exponents. 

Let  ?n,  n  be  any  numbers,  and  a  any  base ; 
then  will  (a™)"  have  every  value  of  a"****. 

(a,  6,  c)  m^  n  commensurables.  [th.4 

{d)  m,  or  n,  or  both,  incommensurables. 

For  let  X,  y  be  commensurable  variables  whose  limits  are  m,  n : 

then         a*=a'*,  [df .  incom.  pVr. 

i.e. ,         every  value  of  a'  =  some  value  of  a*",  [nt.  to  df .  inc.  pwr. 

.-.   (a')*  =  (a'")''  as  x  =  m\ 
but  (a'")>'=(a'")"  as  2^  =  7J,  [df. 

.*.   (a')''  =  (a'")*'  as  x  =  m  and  y  =  n; 
and     •••  A"     =A."*",  [xy  =  mn, 

and     •••   (a')"   equals,  or  includes  among  its  values,  a*^, 

.'.  (a*")"  equals,  or  includes  among  its  values,  a"*",  q.e.d. 
So,  when  only  one  exponent,  m  or  n,  is  incommensurable,  q.e.d. 

Cor.  Wliatever  the  values  of  the  exponents  m,  n,  the  powers 
(a")°  and  (a°)"  have  at  least  one  value  in  comynon,  viz.,  a'"". 

Note.  Most  of  §§2,  5,  with  some  obvious  results,  may  be 
summarized  thus : 

Tlie  values  of  any  commensurable  power  a"'  depend  upon  a 
and  the  value,  not  the  form,  ofn' ;  so  with  any  incommensurable 
power  A°,  =  lim  a°'  as  n  =  n'. 

Any  product  or  quotient  of  like  powers,  whether  in  one  series  or 
not,  is  the  like  power  of  the  product  or  quotient  of  the  bases;  except 
that  if  the  bases  be  not  independent,  the  power  of  the  product  may, 
though  rarely,  have  more  values  than  the  product  of  the  powers. 

Any  product  or  quotient  of  powers  of  one  base,  in  one  series,  is 
that  power  whose  exponent  is  the  sum  or  difference  of  the  given 
exp>onents,  and  is  in  the  same  series. 

Any  power  of  a  power  is  that  power  of  the  ba^e  whose  exponent 

m   p 

is  the  product  of  the  given  exponents;  except  that  (a^)^  may, 

mp 

though  rarely,  have  more  values  than  a^,  ifq  be  not  prime  to  m. 


§  6.]  CONTINUITY   OF  POWEliS  IN   GENEEAL.  209 


§6.     CONTINUITY  OF  POWERS  IN  GENERAL. 

Lemma.    The  positive  value  ofany-l     _^  ^-    power  of  a  post- 
tive  base: 

1.  Is'{   ^^"^j     than  unity  if  the  base  be  larger  than  unity. 

2.  Is-{  .    ,         than  unity  if  the  base  be  smaller  than  unity, 

I.    Let  A,   >  1,  be  an}'  positive  base,  and  n  any^  posi  lye 

laro-er  negative 

exponent;  then  is  the  positive  value  of  a"^       *,,    ,  than  I. 

(a)  n  commensurable.  [th.  5 

(6)   n  incommensurable. 

Let  ?i',  w"be  an}'  two  commensurable  variables,  both  {  ^^^^  ^7^' 

negative, 

approaching  n  as  their  common  limit  in  such  wise  that 
always  n'<»i<n" : 

then    •.•  A  >1    and  n\  n"  are  both  {  ^^^^^^  [hyp. 

.*.  of  A"',  A"",  the  positive  values  both  ^  ^I  ;  [(a) 

and     •.•  the  positive  value  of  A"  lies  between  them,     [th. 8,df. 
.'     the  positive  value  of  a"  ^  ^  1 .  q.e.d. 


2.    LetA<l,   andlet»be^j;°^^«[4=^ 


then   •.•  ->1, 

A 

.-.   ofA%     =(i)'", 

the  positive  value  ^  ^  1  for  -n^  negative,  rj. 

^  '  >  '  positive,  ■- 

n         I  positive. 
I.e.,  for  n<  ^        ..  q.e.d. 

'  '  negative.  ^ 

CoR.    If  A''=l,  then  either  a  =  1  or  n  =  0. 

For  if  neither  a  =  I  nor  n  =  0, 

then         is  a**  larger  or  smaller  than  unity, 

which  is  contrary  to  the  hypothesis  ; 

.♦.  either  a  =  I  or  n  =  0.  q.e.d. 


210  POWERS  AND   ROOTS.  [VIII. 

Theor.  12.    If  tliere  he  a  variable  positive  base  a,   and  a 
constant  -{^        ..     exponent  n,  then: 

1 .  To  each  value  of  the  base  there  corresponds  one  and  but  one 
positive  value  of  the  power;  -{  ^^J^^^^^^^^^^^^  function  of  the  base. 

2.  To  each  positive  value  of  the  power  there  con-esponds  one 

and  but  one  positive  value  of  the  base;   -l  ^^^^^^^^^^"9'  function 

^  ''  ^     '  a  decreasing   ^ 

of  the  x)ov:er. 

3.  TJie  positive  values  of  the  power  and  of  the  base  are  con- 
tinuous functions  of  each  other, 

1.  (a)  n  commensurable.  [th.5,6 
(6)   n  incommensurable. 

For  let  A**  be  the  limit  of  a  series  of  commensurable  powers  of  a  ; 
then    *.•  eachof  these  powers  has  one  and  but  one  positive  value, 
.*.  a"  has  one  and  but  one  positive  value.  q.e.d. 

So,  let  a',  a"  be  any  two  values  of  a,  whereof  a'<  a"  : 
then   •••  a":  a'>1, 

.•.(a":a')"^  ^1;  [lem. 

.-.  a"%   =(a":a')'^-a'«,  {  ^a'", 

e.e.,         the  larger  the  base,  the  ^  1-|- ^  the  {  "^^^  power. 

Q.E.D. 

2.  Conversely : 

•••  a  =  (a'')", 

.'.  to  each  positive  value  of  a"  there  corresponds  one  and 

1    ,  ...  ^        e  (  an  increasina;  /? 

but  one  positive  value  of  a  ;    ^       -,  .    ®  func- 

^  '    '  a  decreasing 

tion  of  A".  Q.E.D.       [1 

3.  Let  A,  always  increasing,  pass  in  order  through  every 

positive  value  from  0  to  +co  : 

■  v.  „      ,  an  increasing  #       +•         «  n 

then   •.*  A",   s      1  •    =*  function  of  A,  \1 

'    '  a  decreasing  '  L 

takes  in  order  every  value 

fromO%   =^  0,tooo%   =  ^  ^,  [2 

.*.  A"  is  a  continuous  function  of  a.  q.e.d.      [df. 

So  A,   =(a")",  is  a  continuous  function  of  a'',     q.e.d. 


§6.]  CONTINUITY  OF   POWERS   IN   GENERAL.  211 

Theor.  13.    If  there  he  a  constant  positive  base  a<{   ^^9  J" 
than  unity ^  and  a  variable  exponent  n  ;  theii : 

1.  To  each  value  of  the  exponent  there  corresponds  one  and 

but  one  positive  value  of  the  power ;  {  ^^(^ecreasiv/^^*^^^^^*^^^  ^^ 
the  exponent. 

2.  To  each  positive  value  of  the  power  there  corresponds  one 
and  but  one  value  of  the  exponent;  ^  „  ^  .^^  •  function  of 
the  power, 

3.  The  exponent  and  the  positive  value  of  the  power  are  con- 
tinuous functions  of  each  other. 

1.  For,  when  the  base  and  exponent  are  given,  there  is  one  and 

but  one  positive  value  of  the  power,  q.e.d.  [th.l2. 1 

And  this  value  is  ^      ,            .  j*  function  of  the  exponent ; 

for  let  Til,  912  be  any  values  of  n  whereof  Wg  >  %  5 

then  •.•  A"2  =  A"2-«i.A"i,                                                           [th.  10 

and  •.*  of  a"2-"i  the  positive  value <(  ^  1,      [lemma;  ng— ni>0 

.-.of  a"!  the  positive  value  \  ^  that  of  aPu  q.e.d.  [II.  ax.16 

2.  (a)  To  each  positive  value  b  of  a"  there  corresponds  one 
value  of  n. 

For  let  b',  b"  be  any  positive  variables  such  that   always 

b'^  ^b^  ^b",  and  approaching  b  as  their  common 

limit ;  and  let  variable  commensurable  exponents 
9i',  n"  be  so  taken  that  always  the  positive  value  of 
A"'  shall  lie  between  b'  and  b,  and  the  positive  value 
of  A""shall  lie  between  b  and  b"  :  [th.  7(2),  cr.l 

then  •••  of  the  variable  commensurable  powers  a"',  a**"  one  in- 
creases and  the  other  decreases  toward  b  as  their 
common  limit,  [t^JP* 

and  the  exponents  n',  ?i"  have  a  common  limit  n  that  lies 

between  them,  [th.7  cr.  3  cnv. 

.'.  the  value  of  this  common  limit  is  a  value  of  the  expo- 
nent n  corresponding  to  the  value  b  of  the  power  a**. 

Q.E.D.     [df.  incom.  pwr. 


212  POWERS   AND   r.OOTS.  [VIII, 

(b)  To  each  positive  value  b  of  a°  there  corresponds  hut  one 
value  of  n. 

For  if  A"*,  A"  each  =  B, 

then    •.*  A"*~''  =  A"*:  a"  =  b:  B  =  1,  [th.lOcr. 

,..  m  — 71  =  0.  [cr.  tolem.  th.  12 
.♦.  m         =n.  Q.E.D. 

(c)  The  exponent  is  ^  ^""^g^rea^^^^^    function  of  the  positive 
value  of  the  power. 

For  this  is  equivalent  to  the  statement,  already  proved,  that 

^.  .^.  ,  -  .,  .    ,  an  increasing 

the  positive  value  of  the  power  is^  ^  decreasing 

function  of  the  exponent.  [1 

3.  For  to  every  vahie  of  the  exponent  there  corresponds 
one  and  but  one  positive  vahie  of  the  power,  and 
conversely;  [1,2 

and     *.*  as  the  exponent  increases  the  positive  value  of  the 

,  alwavs  increases,        .  .  no 

P^^^^-i  always  decreases,  '"''^  ^o-^versely ;         [1,2 

.*.  as  the  exponent  passes  in  order  through  all  values 
from  ~oo  to  +00,  the  positive  value  of  the  power 
passes  in  order  through  all  values  from  {  ^      to    0  ' 

and  conversel}',  as  the  power  passes  from  <|  ^      ,      ^'  the 

exponent  passes  through  all  values  from  ~oo  to  +go  ; 

and  •••  the  power  is  the  limit  of  a  corresponding  commensura- 
ble power  that  changes  infinitesimally  when  the  ex- 
ponent changes  infinitesimally,  and  conversely, 

.♦.  every  infinitesimal  change  in  either  the  exponent  or  the 
power  gives  an  infinitesimal  change  in  the  other, 

.*.  both  exponent  and  power  are  continuous  functions  of 
each  other.  q.e.d.     [df.  contin.  f  unc. 


§  7.]  DERIVATIVE   OF   A   POWER.  213 

§7.     DERIVATIVES   OF  POWERS. 

DERIVATIVE    OF   A    POWER    OF    A    VARIABLE    BASE. 

Theor.  14.    The  derivative  as  to  any  variable  base  of  a  power 

of  that  base  is  the  product  of  the  given  exponent  into  a  power  of 

the  base  whose  exponent  is  a  unit  less  than  the  given  exponent. 

Let  X  be  any  variable  and  n  any  number  ; 

then  will  d^x"  =  n  •  a;""^ 

(a,  6,  c)  n  commensurable.  [VII.  th.  17 

(d)  n  incommensurable. 

For  let  ?i'  be  a  commensurable  variable  independent  of  x  and 

such  that  n'  =  n,  and  let  x  take  any  increment  h  ; 

then   •.•  af ',  a?"  take  the  increments  {x-{-h)'''  —  x"',   (a^+7i)"—  x"", 

and     *.•  a;**'  =  a;'*,  (a;-|-7i)'*'  =  (aj+Zi)",  as  ?^'=  n,  [df.  incom.pwr. 

.*.   («  + /i)"'  — .T"'  =  (aj  +  Zi)**  — a;"*   as    n'  =  n,  whatever  h 

maybe,  [VII.  th.  7 

i.e.,         inca;"' =  inca;" 

■,           inca;'*'  .  incaj'*  /  •        i  n  • 

and  = ,    as    ?i' =  ?i,  however  small  mc  a;, 

inc  X       inc  x 

...  lim^^^  =  lira^-?^^  as   n'^n   and   inca;  =  0, 
inc  a;  inc  a; 

i.e.,          D^x""'       =D^a5".  [df.deriv. 

But     •.•  D,V       =n''X^'-\  [VII.  th.  17 

and     •.*  w'-aj'*'-^  =  ?i-a;"-^   as  n'  =  n,                           [VII.  th. 8 

.-.  D^«"        =n-a;"-^  q.e.d.     [VII.  th.Gcr. 

DERIVATIVE   OF   A   VARIABLE    POWER. 

Theor.  15.  The  derivative  as  to  any  variable  of  that  poiver 
of  a  base  whose  exponent  is  the  variable,  is  the  quotient  of  the 
given  power  by  a  constant  whose  value  depends  upon  the  base 
alone. 

Let  A  be  any  base,  m^  a  certain  function  of  that  base,  x  any 
variable ; 
then  will  d^  a'^  =  a''  :  m^. 

For  let  X  take  any  increment  h ; 
then   •.•  a^+'*-a^=a^.(a'^-1),  [th.lO 


214  POAVEllS  AND   ROOTS.  [VIII. 

A*+*-A'  ,  A*-l 

.'.  D,A*  =A''-lim — ; —  when7i  =  0. 

h 

But  lim^^^=^*^^*~^^       when/i  =  0,  [VIL  th.  18 

h  Dji 

=  DjA*  when  7i  =  0, 

an  expression  free  from  x  and  a  function  of  a  only ; 

and     •.'  a*  has  a  single  value  for  any  one  value  of  7i,  [th.l3(l) 

A*— 1 

,*.  has  a  single  value, 

h 

and  lim ,  when^=0,  has  a  single  value,  dependent 

h 

on  A  alone. 

Put       —  =  lim  — ^^— ,   when  h  =  0, 
Ma  h 

then         D,A'=  A* :  m^.  q.e.d. 

Cor.   Ifebe  such  a  number  that  Me  =  1,  then  Tt^e^  =  e^. 

Note.  The  function  m^  is  called  the  modulus  of  that  system 
of  logarithms  whose  base  is  a  ;  its  value  is  found  by  methods 
in  [XII.  pr.  11].  The  base  e  is  the  base  of  the  Napierian  system 
of  logarithms. 

DERIVATIVE    OF    A    LOGARITHM. 

Theor.  16.  TJie  derivative  as  to  any  variable  of  a  logarithm 
of  that  variable  is  the  quotient  of  the  modulus  of  the  system  by 
the  variable. 

Let  X  be  any  variable,  a  the  base,  and  m^  the  modulus  of  the 
system ; 
then  will  D^log^cc  =  m^  :  x. 

For  put  y  =  logAa;; 
then   '.'  X     =A%  [I.  §11  df. log 

.-.    Dy.T  =  Dj,A''  =  A":  Ma.  [th.  15 

But       *.*    D^2/=  1  •  ^y^ 
=  Ma  :  A% 
.*.  Djloga;  =  M^:  a;.  q.e.d. 

Cor.    DxZo^eX=  1  :  x. 


§  8.]  KADICALS.  215 

§8.    RADICALS. 

A  radical  is  an  indicated  root  of  a  number.  There  may  be  a 
coefficient ;  and  then  the  whole  expression  is  called  a  radical, 
and  the  indicated  root  is  the  radical  factor. 

A  radical  is  <|  .      , .       ,  if  the  root  ^    ^^^     ,   be  found  and 
'  irrational  '  cannot 

exactly  expressed  in  commensurable  real  numbers  [I.  §1],  or 

in  rational  literal  expressions  [I.  §  12"|.    Its  value  is  <(  ^.^^    . 

^  •-  -^  '  imaginary 

.-  .,  ,  do  not  involve  ..  .    ^ 

if  it  ^  .       J  the  even  root  of  a  negative. 

E.g.,    -(/256,  ^8,  ^-8,  ^a^  ^ (a^ ^  2 ab -\- b"")  are  radi- 
cals that  have  the  rational  values 
-2,         2,         -2,       a,         ±(a  +  &), 
besides  certain  irrational  values  discussed  later. 

But       ^x,  Va^  -^a*,  ^a-a''^,  f(a^  + 6^)^  are  irrational, 
and  V-1'  V-«''  </-2a^  |a.(-a)^   |(a  +  &^-l)t 

are  iraaginaries ;  the  first  two  of  them  commensu- 
rable, and  the  others  not. 

An  expression  that  contains  a  radical  is  a  radical  expression. 

A  radical  expression  that  cannot  be  freed  from  roots  is  an 
irrational  expression,  or  surd  [I.  §  12]. 

An  equation  that  contains  surds  is  rationalized  when  it  is  re- 
placed by  an  equivalent  equation  free  from  surds. 

E.g.,    the  equation  x=z  ^2,   i.e.,  x=  ^2  or  ^2, 
when  rationalized,  becomes  a^  =  2. 

Roots  of  rational  bases,  and  integral  powers  of  such  roots, 
with  rational  coefficients,  if  any,  are  simple  radicals  ;  and  a  radi- 
cal is  in  its  simplest  form  when  its  coefficient  is  real  and  entire, 
its  exponent  positive  and  less  than  unity,  its  root-index  the 
smallest  possible,  and  its  base  a  real  and  entire  expression  con- 
taining no  factor  to  a  power  whose  degree  equals  or  exceeds 
the  root-index. 

If  a  simple  radical  be  surd,  it  is  a  simple  surd. 

The  degree  of  a  simple  radical  is  the  value  of  its  root-index. 


216  POWERS  AND   ROOTS.  [VIIL 

A  simple  radical  is  quadratic^  cubic,  quartic  or  biquadratic,  ••• 
when  the  root- index  is  2,  3,  4,  •••. 

E.g.,    |(a2-f^>2)i,    3a62.^(a2-6c^),    a^.a^,    </-3, 

are  simple  quadratic,  cubic,   and  quartic  surds  in 

their  simplest  forms. 
But      ^V«'^    ■</«'»    VS.  V-8,    f(a2c2  +  62^2)^^    ^_5, 

are  simple  radicals  not  in  their  simplest  forms ;  for 

they  may  be  severally  reduced  to : 

"^a^a,    a^a,    2^2,    2V~2,    ^c- {a" -^W)^, 
li-^245,  =  11.2451 

And     V[2-v/(3+</4)],   (a^+^^')%   {a^-b^-iy, 
are  complex  radicals  or  surds. 

Two  radicals  are  like,  or  similar,  if  they  have  the  same  radical 

factor  \   ■{  ^         J.7    if  they  •{  .be  made   like    by 

'    '  non-conformable  ^  '  cannot  ^ 

transformation. 

E.g.,    3a^    -b^a-,    2x' {a" -^W)^,   -4?/ .  (a^  +  ft^)^, 
8  (a  —  ?/)  •  {a?  +  &-)  ^,    are  like  radicals, 
and  V^S'    V^2»    V^S'         ^^®  conformable. 

The  sum  of  two  non-conformable  simple  surds,  or  of  a  rational 
expression  and  a  simple  surd,  is  a  binomial  surd;  the  sum  of 
three  non-conformable  simple  surds,  or  of  two  such  surds  and  a 
rational  expression,  is  a  trinomial  surd;  and  so  on. 

Two  quadratic  binomial  surds  are  conjugate  if  they  differ  only 
in  the  sign  of  one  teim. 

E.g.,    a  +  V^  a-V^;     10^  +  3,   10^-3; 
V-'^'  +  V(2/  +  ^)5   V^-V(y  +  2;). 

Two  surds  are  complementary  if  their  product  be  rational. 

E.g.,    a*,  a*;    ffl,  5"^    V(«'  +  6'),  ■^/{a'  +  b')', 

a+V^'  a— V^5    (^-\-b-y/—\,  a  —  b-y/—l. 

So,       any  two  conjugate  binomial  surds  are  complementary. 

E.g.,    a-\-^b,a-^b',     2  +  3V1,  2-3V1- 


§8.]  BADICALS.  217 

Theor.  17.    If  two  simple  surds  in   their  simplest  form  he 
equals  their  coefficients  are  equal  and  their  radical  parts  are  equal. 

Let  a  ^A,  h  y\^  be  equal  simple  surds  in  their  simplest  form  ; 
then  will  a  =  &,    a  =  3,   m^n. 

For  let  m  =fp^   n  =fq, 
wherein  /is  the  h.c.msr.  o/m,  n,    and  j5  is  prime  to  q  ; 
then   •.•  a.^A     =b'^B,  [hyp. 

.-.  a^^.A      =6/^.(^b)^  [II.ax.6 

=  b^^'(  -^bY  ;  [df ,  f ract.  pwr. ,  th.  1 

.-.  a^^A:6^^  =  (^B)^ 

a  true  equation,  but  true  only  when  (-^bY  is  rational, 
i.e.,  when  p  =  q   and   m  =  n;  q.e.d. 

.-.  a:6=  V(R:  a),  [th.2  cr.  1 

a  true  equation,  but  true  only  when  ^(b  :  a)  is  rational, 
i.e.,  when   a  =  b   and    a  =  b.  q.e.d. 

Cor.  1.    Two  non-conformable  surds  cannot  be  equal. 

CoR.  2.    The  product   or  quotient  of  two  \  ^         t  7 

^  ^  ./  J  non-conformable 

,  -,     ..  1    •    )  rational. 

simple  quadratic  surds  is  -{         -. 

E.g.,    ^C)  is  conformable  with  -y^f  but  not  with  -^5, 

and  V(^'t)'  V(^  •  J)'  V(f  •  ^)   ^^®  *^®  rationals 

±2,  ±3,  ±i, 

but  y'(6-5),  ^^(6:5),  -y/ (5  ;  J),  etc.,  are  surds. 

Cor.  3.    If  the  continued  product  or  quotient  of  two  or  more 

simple  quadratic  surds  be-{         ■.      '  theri  the  continued  product 

or  quotient  of  any  of  them,  and  the  continued  product  or  quotient 

of  the  rest  of  them,  are  {       ^       ^      *  ,, 
-'  J  ■>         \  non-conformable. 

E.g.,    V2-V3-V6  =  6;    V^  •  V^  *  V^  =  V^O, 

and  V(2-3),  V^  5  V(2  -  3),  V^  5  -•  ^^'^  conformable  ; 

but  -^(2 '3),  -y/5  ;  -y/(2  :  3),-y/5  ;  •••  are  non-conformable. 


218  POWERS  AND  KOOTS.  [VIII.  th. 

Theor.  18.    Tlie  sum  of  a  finite  number  of  simple  non-con- 
formable surds  cannot  be  rational. 

Let  Oi^Ai,  a2^A2,  •••  CThV^'   =  </'»!'  ^^2,  •••  V^n,  ^G  any 
simple  non-conformable  «urds  ;  and  let  c,  as  well  as 
«i)  A-i,  Bi,  •••  a^,. A„,  B„,  be  rational: 
then         the  relation  -^Bi  +  V^2  H-  •••  +  V^»  ==  c  is  impossible. 

(a)  One  swrd,  c  ^  0  ;  or  two  surds,  c  =  0  ; 

i.e.,  -^Bi  =  c  is  impossible,  q.e.d.     [df.  surd 

and  -^Bi  -f  ^B2  =  0  is  impossible.  q.e.d.  [th.l7  cr.l 

(b)  Two  quadratic  surds. 

If  possible,  let  V^i  +  V^-'  =  ^> 
then         Bi  H-  2 ^Bi  62  +  62  =  c^ 
.-.  2 Vbi62=c-  — Bi  — B2, 
i.e.,  a  surd  equals  a  rational  number,  [th.l7cr.2 

which  is  impossible  ;  [df.  surd 

.-.    ^Bi-\-^B2=^C.  Q.E.D. 

(c)  TJiree  quadratic  surds. 

If  possible,  let  -y/^i  +  VB2  +  V^s  =  ^  5 
then   •.•  Vb2  +  Vi^3  =  c— V^i'  l^yV- 

.-.    B2  +  2VB2B3+B3  =  C-  —  2cVBi  +  Bi, 

.-.   2cVbi  +  2V32B3  =  C^  -f  Bi  —  B2  —  B3. 
So,       2cVb2  +  2Vb3Bi  =  c2  +  B2  — B3  — Bi; 

2  c  VB3  +  2  VB1B2  =  c^  +  B3  —  Bi  —  B2 ; 
i.e.,  the  sum  of  two  non-conformable  surds  is  rational, 

which  is  impossible  ;  [(6) 

or  else  -^Bi  is  conformable  to  VB2B3,  V^a  to  VB3B1,  VB^to  VB1B2? 
and  c^+Bj— Bo—  B3  =  C-+  B2— B3—  Bi  =  0^+63—  Bi—  B2  =  0, 

whence    Bi  =  B2  =  B3, 
and  Vbi  ±  a/bi  ±  Vbi  =  c, 

which  is  also  impossible  ;  [(a) 

.-.     VBi+V^2+  V^ST^C-  Q.E.D. 


18,  §  8.]  RADICALS.  219 

(d)  Any  number  n  of  quadratic  surds,  c  =  0. 
1.    The  assumed  surd  equations 

Vbi+ v^2+ Vb3=  0,  Vi5i+ Vi^2+ V%+ V^4=  0, ..., 

VBi+VI52+VB3H h  V»n=  0 

may  be  reduced  to  the  equivalent  surd  equations 

Kg  =83^^2153,     R4  =  S4V^3B4?     --"J    Rn=SnVl^n-lBn5 

and  to  the  rational  equations 

T3=0,     T4  =  0,      ...,     T„  =  0, 

wherein   R3,  S3,  T3  ai^e  rational  functions  of  Bi,  B2,  Bg ; 

R4,  S4,T4,  0/  Bi,  .•.,B4;     ...  ;     ^,8n^T^,  of  Bi,"'B^. 

For  if  Vbi  +  \/^2  +  V^3  =  0, 
theu   •.•  V^^i  =  —  V^2  —  V^3» 

.-.    Bi  — B2  — B3=2  VB2B3J 
i.e.,  K3=  S3-y/B2B3j  q.e.d. 

and  (bi  —  B2  —  B3)^  —  4B2B3  =  0, 

i.e.,  T3  =  0.  Q.E.D. 

So,  in  the  last  two  equations  replace  -y/B^  by  y'Bg  +  y'B4 ; 
then   •.•   [bi  — B2  — (VB3+ Vb4)^^  — 4b2(Vb3  + Vb4)^  =  0, 

.-.    2Bi2_22BiB2  +  8B3B4  =  4(Bi  +  B2-B3-B4)VB3B4, 
i.e.,  R4  =  S4V^3^45  Q.E.D. 

and  R4^  —  84^6364  =  0, 

i.e.,  T4=0.  Q.E.D. 

So,  if  the  law  holds  true  for  7c  surds,  it  holds  true  for  A; + 1  surds. 

Fot  in  the  equation  T;fc=  0  replace  ^B;^  by  ^b„  +  Vb*+h 
i.e.,  replace  B;^  by  b,  +  b,+i  +  2  Vb^^b.+i  ; 

then   •.•  T;^  =  0  becomes  R;t+i  =  S;t+i ^B^B;t^.i,  q.e.d. 

.*•    ^"k+l —  S  t^iB^Bj^l  =  (J, 
i.e.,  T;i.^.i  =  0.  Q.E.D. 

But  •.•  the  law  holds  true  for  3  surds  and  for  4  surds, 

.'.it  holds  true  for  5  surds,  for  6  surds, ...  for  w  surds. 

Q.E.D. 


220  POWERS   AXD   EOOTS.  [VIII.  th. 

2.   The  assumed  surd  equations 

Rg  =  S3  VB2  B3,     R4  =  S4  V^3  B4^      •  •  •  Rn  =  Sn  V^n-l  Bq 

are  a?Z  impossible. 

For     *.*  V^i?  V^a?  V^s?  *••  V^n  are  non-confonnable  surds, 
.-.   VB2B3,  V^sB**  •••  VBn-iB«  are  surds  ;  [th.l7  cr.  2 

.-.  in  each  of  these  assumed  equations  a  rational  number 
stands  equal  to  a  surd,  which  is  impossible,  [df.  surd 

or  else     r„,  r„',  Rn",  •••,  s«,  s„',  sj',  .••,  all  =  0, 

wherein  r„',  •••are  what r^,s„ become  when  Bi,--- B„are permuted; 

e.g. ,         S4,  84',  S4"  are  Bi  +  Bg  —  B3  —  B4,  b^  4-  B3  —  B2  —  B4, 

Bj  +  B4        B2        63! 

and  if  R^,  R„',  Rn",  •••»  s„,  s„',  s„",  •••  all  =  0, 

then         Bi  =  B2=  •••  =  B„, 

and  Vbi»  •  *  •  "v/Bn  are  conformable,  which  is  impossible,  [hyp. 

.*.     VBi+  VB2+VB3t^0,    VBi+VB24-VB3+VB4=^0,-.-, 
VBi+  VBgH h  VBn=^0.  Q.E.D. 

(e)  Any  number  of  quadratic  surds^  c  =?t:  0. 
Take  V^n+i  a  simple  surd,  and  Di= Bi-d„+i  :  c^  •  •  •  d„=b„-d„+i :  c^; 
then    •.•  Vi>i  +  ---  +  Vi>«=?^  V»«+i»  [W 

.-.  VbiH f-VBn=?^c.       Q.E.D.     [mult,  by  (c  :  V»n+i) 

(/)  Any  number  n,  of  surds  not  all  quadratic,  c  =  0. 
1.    TJie  assumed  surd  equations 

^Bi  +  VB2  +  VB3  =  0,     VBi+  VB2+  VB3+  ■^B4  =  0, 
VBi  +  ^B2  +  VB3  +  -  H-  >/Bn  =  0, 

wherein  each  simple  surd  is  in  its  lowest  terms,  may  be  reduced 
respectively  to  the  equivalent  surd  equations 

R3=S3«V3,     R4=S4'V4,      •••,     Rq  =  Sq  •  Vq, 

and  to  the  equivalent  rational  equations 

Ts=0,     T4=0,      ...,    Tn=0, 

wherein  R3, 83,73  ai^e  rational  functions  of  Bi,  B2,  B3 ;  ••• ; 

R45  S4,  T4,  0/   Bj,  •••,  B4  ;    •••,    Rn)  Sq,  Tq,  0/  Bj,  •••,  Bq. 

1      1 

and  V3  =  the  surd  B2  "^  •  Bg',  =  ^b^'^'^z^' , 

1      1 

V4  =  the  surd  Bg  '^  •  84%  =  -Vbs""  "  64'"",  •  •  • , 

and  h  =l.c.  mlt.  0/  q,  r  ;  q',  r'  =  the  integers  h  :  r,  h  :  q  ; 

k  =l.c.  mlt.  0/  r,  s  ;  r",  s"=  the  integers  k  :  s,  k  :  r; .... 


18,  §  8.]  KADICALS.  221 

For  in  the  assumed  surd  equation  -^Bj  +  ^Bg  =  0,  and  in  the 
equivalent  rational  equation  b^'  =  ±  Bg^,  replace  Bj 

then         Bi'  =  ±B/(l+V3)^S 

•    .-.  B/=  ±  B/[l  +  i)gY3+  ^^'^^^  ~  ^^  V3^  +  ...]'  [bin.th. 

But  this  equation  can  contain  not  more  than  h—1  surds  ; 
for  if  Vg*,  V3*+\  V3*+^,  •••be  present  they  are  conformable 

to  V3°,  Ys\  VsS  •••; 
.*.  the  equation,  reversed,  reduces  to  the  form 

V3'-'  +  AiV3*-2  +  A2V3*-«+...+A;,_2-V3  +  A;,_l  =  0, 

wherein       Aj,  Ag,  A3,  •••  Aa_i  are  rational. 

Letx  =  V3*-iH f-Aft+i;    w  =  V3*-B2-'"'b3'',  =  0.        [df.Va 

Divide  w  by  x  :  the  remainder  y  has  no  power  of  Vg  above  y/'^. 
So,  divide  x  by  y  :  the  rem'der  z  has  no  pwr.  of  V3  above  Vs*"^;  •  •  • ; 
and     •.*   -y''b2~'"b3''  is  a  simple  surd  in  its  simplest  form, 

.-.  V3*— Bg^'^'Bg'',  or  w,  has  no  rational  factor,  [df.  sim.  form 

.*.  w,  X  have  no  rational  common  factor ; 

.'.  the  divisions  go  on  till  a  remainder  is  reached  having 

only  the  first  power  of  V3 ;  and  then,  one  free  from  V3. 

Let  R3— S3  V3,  T3= these  remainders, wherein  Kg,  S3,  Tgare  rational; 

then   *.•  w  =  0,   x  =  0,     .•.  each  successive  remainder  is  0, 

i.e.,  R3—  S3V3=0,     T3  =  0.  Q.E.D. 

So,  in  the  assumed  surd  equation  V^i  + -^^Ba  +  ■{/B3  =  0 ,  and 
the  equivalent  rational  equation  T3  =  0,  replace  B3  by 
(Vb3+Vb4)%   =B3(14-V4)^ 
then         the  surd  equation  -^Bi  +  -^ Bg + ^^3 + VB4 = 0  is  equiva- 
lent to  an  equation  x'  =  0  with  no  surds  but  V4,  •  •  •  v/"^. 
Let  w'  =  v/  —  Bg"*"^ B/" ;  and  divide  w'  by  x',  x'  by  y',  ••• ; 
then         the  final  remainders  give  R4—  S4V4=  0,  t^=  0.  q.e.d. 
So  for  any  number  of  surds.  q.e.d. 

2.    The  equations  R3=  SgVg,  R4  =  S4V4,  •••  are  all  impossible. 
For      Kg,  R4,  •••  are  rational,  and  S3V3,  S4V4,  •••  are  surds. 


222  POWERS   AND  ROOTS.  [VIII.  prs. 

{g)  Any  number  of  surds  not  all  quadratic^  c  ^^  0. 

For,  if  possible,  let  ^Bj  -f  -^Bo  -\ \-  ^b„  =  c  ;  and  multiply 

by  "x/b^^i,  any  surd  non-conformable  to  the  others  ; 
then         ^B„+i .  -^Bi  +  —  +  ^/B„+i .  S/B„  -  ^B„+i .  c  =  0 , 
wherein  each  term  may  reduce  to  a  simple  surd. 

But  this  last  equation  is  impossible  ;  [(/) 

.*.  the  given  equation  is  impossible. 

Note.  From  principles  developed  in  X.,  XIII.,  it  would 
appear  that  t„,  with  perhaps  a  numerical  coefficient,  is  the  con- 
tinued product  of  some  or  all  of  the  pqr  •  •  •  v  different  values 

of  the  polynomial  ^Bi  -\ h  V^»  g^*  ^y  combining  each  of  the 

p  values  of  ^Bj  with  each  of  the  q  values  of  -^Bg,  •••. 

E.g. ,  if  V^i  +  V^2  +  V^3  —^    ^^  assumed  true, 
then         T3  =  ( V  Bi  +  V^z  +  V^s)  •  (  V^i  +  V^a  +  V^s) 

•(VBl+VB2+VB3)-(VBl+VB2+VB3)=0.[(d) 

CoR.  1.  If  two  irreducible  polynomial  surds  be  equal,  each 
simple  surd  in  one  j)olynomial  equals  a  simple  surd  in  the  other 
polynomial;  and  the  rational  terms,  if  any,  are  equal. 

CoR.  2.  A  simple  surd  cannot  be  the  sum  of  a  rational  number 
and  a  simple  surd,  nor  of  two  simple  surds.  [(6,  d) 

CoR.  3.  If  A,  B,  a,  b,  a',  b',  •••  be  rational;  ^c  a  quadratic 
surd  ;  m,  n  integers;  and  f  any  rational  function  with  no  irra- 
tional coefficients,  then  : 

(a)  TTTien  A-f  bVc=  (a+bVc)  ^  ^(a'+b'Vc), 

then  A  — b-y/c=  (a  — b^c)  i  X  (^'~^'•^/<^)• 

(&)   WJien  A -\-By/c  =  F (sL-\-h^c,  a'-f  b'Vc»  '"), 
then  A  — B-y/c  =  F(a  — b-^c,  a'— b'^c,  •••). 

m 

(c)    TTTie?!  A-f-B-^c=  (a4-bVc)°j 

m 

then  A  — By'c=  (a  — b^c)°. 

The  reader  may  prove  (a)  by  performing  the  indicated  opera- 
tions ;  (6)  by  means  of  (a)  ;  and  (c)  by  the  binomial  theorem, 
first  raising  each  member  to  the  nth  power. 


1-4,  §  9.]  OPERATIONS   ON  r.ADICALS.  223 

§  9.     OPERATIONS    ON    EADICALS. 
PrOB.   1.       To    REDUCE    A    RADICAL    TO    ITS    SIMPLEST    FORM. 

Resolte  the  number  whose  root  is  sought  into  two  factors, 
whereof  one  is  the  highest  possible  perfect  p)ower  of  the  same 
degree  as  the  radical^  and  the  other  is  an  entire  number ;  ivrite 
the  root  of  the  first-named  factor  as  a  coefficient  before  the  indi- 
cated root  of  the  other.  [ths.  2, 9;  3, 10;  4, 11 

E.g.,    -^48a3&4=^(8a36-^66)-..  =  2a5^6  6. 

So,       -;y(a"6'"  — a'*c^)=  -;;/[a'*.(6'"  — c^)]  =  a-;y(&'"  — c^). 

Prob.  2.    Inversely,  to  free  a  radical  from  coefficients. 

Raise  the  coefficient  to  a  power  whose  exponent  is  the  root- 
index  of  the  radical;  multiply  this  power  by  the  expression  under 
the  radical  sign,  and  put  the  same  radical  sign  over  the  product. 

E.g.,    2a6^66=^(8a-^63.66)=^48a^6^ 

So,       a-yC^"*  — c^)=  V[^*"*(^"*  — c'')]=  y{oI'h"'  —  or'G'')' 

Prob.  3.     To  reduce  radicals  to  the  same  degree. 

Write  the  radicals  as  fractional  powers;  reduce  the  fractional 
eocponents  to  equivalent  fractions  having  a  common  denominator, 
restore  the  radical  signs  using  the  common  denominator  as  the 
common  root-index  and  the  new  numerators  as  exponents,     [th.  1 

E.g.,    ax,-^by,^{b-\-c)==ax,  (by)^,       (b  +  c)^ 

=  (ax)^^,         (by)^\     (b  +  c)  ^^ 
=  ^(ax)"",  ^{byy\  "^{b  +  cy. 
Prob.  4.    To  add  (or  subtract)  radicals. 
Reduce  the  radicals  to  their  simplest  form ;  add  {or  subtract) 
like  radicals  by  prefixing  the  sum  {or  difference)  of  their  coeffi- 
cients to  the  common  radical  factor;  write  unlike  radicals  in  any 
convenient  order.  [II.  prs.  1 , 2 

E.g.,    3V8  +  5V2-10V32  =  6V2  +  5V2-40V2 

=  -29V2. 
So,       a^b  +  a'-^b^  -  a^  ^b'  =  a^b -\-d'b-^b-a^b''  -^b 

=  {a-\-a'b-a^b')^b. 


224  POWERS  AND  HOOTS.  [VIII.  prs. 

PROB.   5.      To    MULTIPLY   (OR   DIVIDE)    RADICALS. 

Reduce  the  radicals  to  the  same  degree;  to  the  product  {or 
quotient)  of  the  coefficients  annex  the  product  {or  quotient)  of  the 
radicals.  [ths.2*,9  ;  3, 10 

E.g.,    3V8-5V2--10V32  =  -3.5.10.V(S-2-32) 

=  - 150 .  V512  =  -  2400 .  V2. 

So,       ab^-a^b^  :  a^b~^  =  a^+2-3 . ftH I  + 1  ^  54^ 

PrOB.   6.      To    GET   A    POWER   (OR   ROOT)   OF    A    RADICAL. 

Multiply  the  exponent  of  the  given  radical  by  the  exponent  of 
the  power  sought.  [th.  4,11 

E.g.,    (3.8^)3      =27.8'       =432.2^  =  432V2. 
So,        ^(3.VS)=^/(V'2)=-^72; 

(a« .  v^')'  =  «"  •  V^""  =  «''  •  ^"  •  V^ ; 

{a^'b^)^     =a^6TV. 

PrOB.  7.  To  REDDCE  A  FRACTION  WITH  A  SURD  DENOMINATOR 
TO   AN    EQUIVALENT    FRACTION    WITH    A    RATIONAL    DENOMINATOR. 

(a)  Tlie  denominator  a  monomial :  Multiply  both  terms  of  the 
fraxAion  by  some  complement  of  the  denominator.  [§  8,  df.  comp. 

(6)  Tlie  denominator  a  simple  binomial  quadratic  surd:  Mul- 
tiply both  terms  of  the  fraction  by  the  conjugate  of  the  denomi- 
nator. [§8,df.  conjg. 

(c)  Tlie  denominator  a  binomial  quadratic  surd  containing  a 
complex  radical :  Multiply  both  terms  of  the  fraction  by  a  group 
of  conjugate  radicals  that,  taken  together,  are  complementary  to 
the  denominator. 

En       ^  =  ^ •  ^^  .  g        ^ot-(V&  +  Vc). 

5f         b     '   V^-Vc  h-c 

a  ^«-V(&  +  Vc)^a-(6-Vc).V(6  +  Vc). 

a  ^<^-[^+V(c  +  V^)l 

&-V(c  +  V^)  b'^-c-^d 

{b^-cY-d 


5-8,  §9.]  OPERATIONS    ON   EADICALS.  225 

(d)  TJie  denominator  any  Mnomial  surd:  Multiply  the  two 
fractional  exponents  of  the  denominator  by  the  l.c.mlt.  of  their  de- 
nominators^ and  attach  the  products  as  exponents  to  the  two  bases; 
divide  the  sum  (or  differeyice)  of  the  powers  so  found  by  the  de- 
nominator^ and  multiply  both  terms  of  the  fraction  by  the  quotient. 

E.g.^  to  rationalize  the  fraction  ; 

23-  H-  3^ 

then   •.•  12  is  the  1.  emit,  of  3,4,    and  12.(|,  f)  =  8,  9. 
and     •.•   (2«-3^):  (2^  +  3') 

=  2^-2^.3^  +  2^.3^-... +  2^.3^-3^, 
6^       _6^    2^-2^. 3^  +  ...  +  2^. 3^-3^. 


2^  +  3^  2«-3« 

PrOB.  8.     To    FIND  A  SQUARE    ROOT   OF    A   BINOMIAL   QUADRATIC 
SURD. 

Let  a  +  ^b  be  a  binomial  surd,  and  x  -j-  -y/y  =  ■y/{a-\-  -yjb) , 
wherein   a,  -^y  are  to  be  found. 

Square  both  members  of  this  equation ; 
then    •.*  x^-\-y  -\-2x^y  =  a-\-  ^b^  (1) 

.-.  :x?-{-y  =  a,    2x^y=^b.  [th.l8cr.l 

Subtract  the  second  of  these  equations  from  the  first ; 
then         a^  -{-y  —  2x^y  =  a  —  ^b.  (2) 

Add  equations  (1,  2)  together,  and  multiply  them  together; 
then         x^-\-yz=a,   and   x^ —  y  =  -y/(a^ —  b). 

Add  these  two  equations,  divide  by  2,  and  take  the  square  root ; 


then 
So 


2 

2  ' 


v^=V- 


la+V(a'-6)   ^      |a-V(a^-d) 


and  x  —  ^y,  =y/{a  —  ^V), 

=   >+V(«'-5)  _  Ja-V(a'-6) _  ^^^_ 


226  POWERS  AND  ROOTS.  [VIII.  pr.  9 

Note.    Sometimes  a  square  root  of  a  surd  of  the  form 
a  +  ^h  -\-  y'c  +  V<^    iii^y  ^G  found. 

Write  V^  +  V^+V2^=V(«+V^  +  Vc  +  V^)» 
then         x+y-\-z-\-2-yJxy+2^xz-\-2y/yz=a-\--yJh  +  y/c+^d. 

Write  x-{-y+z=a,    2^xy  =  ^b,    2^xz=^c,    2^yz  =  -sjd, 
and  find  vahies  for  x^  y^  z  that  satisfy  these  equations. 

E.g.,  to  find  a  square  root  of    9  +  2 yS  +  2-yJh  +  2^\h. 
Write  a; +  2/ +  2  =  9,  2Va;?/  =  2V3,  2^xz  =  2y/b, 
2yjyz  =  2V15. 
then         a;=l,    2/  =  3i    2  =  5,    and  the  root  sought  is 
1+V3+V5. 

PrOB.  9.      To   FIND   A   CUBE  ROOT   OF   A  BINOMIAL   SURD. 


3/ 


Let  a  +  V^  ^®  ^  binomial  surd,  and  x-\--yJy  =  V a  -j-  -y/b, 
wherein   x,  ^y  are  to  be  found. 

Cube  both  members  of  this  equation ; 
then   •.•  a^  +  3a;y  +  (3ar^  +  2/) Vy  =  ^  + V^» 

.-.  ic»  +  3a^=a,    (3a^  +  2/)V2/  =  V^;  [th.l8,cr.l 

and  o?-\-^xy  —  {^x^-{-y)^y  =  a—^h., 

i.e.,  (x-^yy=      a-^b, 

.-.  x  —  ^y       =-Va  —  ^b. 

But      x-\-^y       =  -^a  +  -yjb.  [  hyp. 

Multiply  these  last  two  equations  together ; 
then   •••  ocr  —  y=-\/a^  —  b  =  m,  say, 
.-.  y         =a^  —  m. 

Replace  y  hy  x^  —  m  in  the  equation  ar^  +  3  a;?/  =  a  ; 
then         a? -\-^x{x^ —  m)  =a, 
i.e.,  4a^  — 3??ia;  =  a. 

From  this  point  on  there  is  no  general  solution,  but  particular 
examples  may  be  solved  by  finding  a  value  of  x  by  inspection 
from  the  equation  4a;^  —  3  mx  —  a. 

E.g.,  to  find  the  cube  root  of  10  +  6  -^3  ; 
then         a  =10,  6=108,  m=  ^(100 -108)  =  - 2  ; 
.-.  4a^+6a;=10, 
.-.  a;=l,    2/ =  3,    and    1  +  ^^  is  the  root  sought. 


§  10.]  EXAMPLES.  227 

§  10.    EXAMPLES. 

§1. 

1 .  Replace  the  radical  signs  by  fractional  exponents  in : 

V«^  </^'^  ■^^;  -</{^'^f)-,  </{b'  +  ^-hy'). 

2.  Replace  the  fractional  exponents  by  radical  signs  in : 

§  2,  THEORS.  2,  3,  4. 

•  ••11.    Multiply  or  divide  as  indicated  : 

3.  (21a)^:(26)*;    (x+S)^.  (a;- 3)^  ;    {20  ab)^  -  (5  ac)^ . 

4.  (2abc)^-(Sacd)^-{6bd)^;   2(abc)^ '-S(a'c)^  : -4.{b^c)K 

1  1  1  1  11 

5.  (a  +  6)^.(a  +  6)".(a-6)-.(a-6)».(a2  +  62)n.+n^ 

6.  ah  a;"^  ;  a^  •  a~^  •  a~^  :  a"^  ;  a^-a'^i  (a^^  •  a^  •  a"  *^) . 

(-3a-'-\-2a-''b-')'{-2a-^-Sa-'b). 

9.  (a^  -  a^  +  1  -  a"^  4-  a"^)  •  (a*  +  1  +  a~^) .  (a*  —  a"^)  •  a^. 

10.  (a;-2/):(aj-'-2/~^);    {x' -  y')  :  (x^  -  y~h - 

11.  (2a^2/"^-5a^?/-2+7a^2/-i-5£c2-|-2a;?/):(a^2/-3-a^2/~^4-iC2/-^). 

12.  Simplify  the  fraction  : 

a^  —  ax~^  +  a^a;~^  —  x~^ 

a^  —  a^a;~  ^  _|_  a^a?"^  —  ax"  ^  _j-  a-*  cc~^  —  a;~  ^ 

13.  Get  the  square  of  : 

(a^-fts)*.    Sah-^x^;   x-{ay)^\   ^a^x'^ -2a~^x^. 

14.  Get  the  cube  of  : 

2(3a)*;   3  a*  6"  ^  a;?/"' ;    (a^- &*)^  ;  a;+ V2/;   a"^-a;i 

15.  Express  in  simplest  form  : 


f21a\-l       r/a-2m\_2-|2„       r2/; 


3^^\r 
4    2/ 


1* 


228  POWEES   AND   ROOTS.  [VIII. 

16.  Get  the  product  of: 

17.  Get  the  square  of  : 

2g^x4-3gx*-2g2x~^-3g^a;-^    d'^b-\-a-^b''a~'''h\ 


¥■■  (f)'+©*H5)' 


«_2(^^Y+3-2/^^V^  +  (^^^''-   ^^'^^  '  ^^^  '  ^^'^^ 


18.  Get  the  cube  of : 

2jy-l^sx-^y^;   ja-f_4x-^gh   ia~h~''^ -\- 5a^bK 

19.  Find  the  square  root  of : 

a;-2_  6x-'y-^  +  92/"' ;   g"^-  4g-^  -  2g-'*4-  12g-3+  9g-2 ; 

4g-i-12g~^6^  +  96^4.i6a~^ci-246^c*4-16c^; 

20.  Find  the  cube  root  of  : 

ig«-f  g26^+6a6-86^  ;  ^x-^^\x^y^-h^x^y'+-,i-,yK 
^a-i-  27x-^a'  -  {x^a)''^  +  {Sx-^a^y. 

21.  Find  the  fourth  root  of  : 

^a^ -  Ax^y-^^^x^y~^  -  2b0x'^y~^^'  +  625y~^'. 

§  5,    THEORS.  9,  10,  11. 

...  25.   Multiply  or  divide  aa  indicated  : 

22..' a;^'./':^^';    (24g)^3:  (66)^3  .    ^^^sy-(x-3)\ 

23.  a.^^^a.^^^:.^^^    J-^.J-^:J;    {a''-b^')-(a^'  +  b^'). 

24.  10^-^i°2- .  lO--^^-,  =  20 .  500  ;'  lO^-^^-  :  lO^-^'"'^-,  z=500  :  20. 

25.  (/^-/^):(a.v^i-2/V^^);  [a^V2(g5)^^H-6^'] :(g^*+6^^). 
•  ••28.    Get  the  powers  and  roots  as  indicated  : 

26.  (lO^-^i^-)',  =20^;  ^(10-^103)1^  ^^2;    (lO"'')'^';     (lO")". 

27.  {2a/"^x-j-Sax'^^y;  (4a^Vl2a^*a.'^^-f-9/')^;  (x^'^y^yK 

28.  {ix^k^^yrty.   (g^'-3g2v^H^*+3g^352^*-6^')i 


§  10.]  EXAMPLES.  229 

k 

§  7,    THEORS.  14,  15,  16. 

In  the  following  examples  e  is  assumed  to  be  such  a  number 
that  Me  =  1 ,  and  a  is  any  constant. 

•  ••32.    Find  the  derivatives  as  to  x  of  the  variable  powers  : 

11       1      „i.       _l      _i 

29.  e'';  a'';    of;  e"* ;  a~*;  a;-*;  e';  a^;  af;  e"*;  a"*;  oT^. 

30.  e^'';   e«  ;   a^'' ;   a«*;   e^'';   a^'^;   a;«*;   a;«* ;   aJ«'^ 

31.  e^(l— a^;    e(«+^)' ;  e-«'^;  av^C^'--^-)  ;   a-v^(«'-^'). 

•  ••  35.    Find  the  derivatives  as  to  x  of  the  logarithms  : 

33.  log,e^;    log„a";    log^af;   log.e-'";    log^cr^;    log^a;-^ 

34.  log,  (a  +  bx»)  ;    log,  [log,(a  +  &aj")]  ;    log,  (e==  -  e'^). 

§  9i    PROB.  1. 

•  • .  39 .   Reduce  to  simplest  form  : 

36.  125^;  567^;  392^;  1008^;  216*;  72*;  162*;  48^;  160*. 

37.  (llJ-f)*  (6|i)*;   (101)^;   (eff)-!;  2500^;    ^296352. 

38.  ^Ulx-^yz";    ^56a'b'c^;    ^112a-^6-V;    ■Sy64:a'b-^G\ 

39.  V(72a'6-726+18a-26);    ^[x'y-'^-xy^ -3af(x-y)^. 

§  9,   PROB.  2. 

•  ••43.    Free  from  coefficients  : 

40.  6V5;    2V^;    2a;V2;    ^a^5b;    4-^6;    5a^2/;    IV^J- 

41.  iV2&;   5Vic;   27a';   ^VCI)*;   *(f)^^  t^r^'d 2/^-^)*^ 

42.  3a2^2a2d2;  i^4a^.y;  5a-^j\ay',  iabc-^Sa'b. 

§  9,  PROB.  3. 

•  ••45.   Reduce  to  the  same  degree  : 

44.  a^  a*;    a*,  6^;    3*,  4^;    ^ab,  -*/ac,  ^bc,  -^(b  +  c), 

45.  a*,  6^  ;   a^,  b^,  c^  ;  a;*,  a;^,  a;^,  x^,  x^  ;    (3  a;)*,  2?/^,  4  A 


230  POWERS  AXD   ROOTS.  [VIII. 

46.  Which  is  the  greater :    (i)^or(|)^?    ^2  or -^3? 

-^9or</18?   m^  or  (m  +  1)^  whenm>3? 

§  9,   PROB.  4, 

•  ••51.   Add  of  subtract  as  indicated  : 

47.  V18-V8;    V12S-2V50  +  7V72;    6Vf-3VI- 

48.  9V80-2V125-5V245+V320;    3VI  +  4VtV- 

49.  625* -7. 135^ +  8.  320*;   8.(|)*  +  ^  •  12*  -  |.27^. 

50.  6(8a«6)*+4a(aS6*)^-125(a«60*;   a^b^ -\-2ab^ -{-bK 

51.  2^i+3^^;  |(|H)*-V(HI)^;  f«2>*-l(2>  =  0*. 

§  9,  PROB.  5. 

•  ••59.   Multiply  or  divide  as  indicated : 

52.  3V2-2V3;    8V6:2V2;   5V7-2V7;   3^^2*. 

53.  3V6-2V3-4V5:12V10;    4  V3-3  V5-5-^2  ;    2*^3*.4^. 

54.  A-(l)*^(f)-(il)^;   5^.4^.3^60^   (f)^:(f)^^ 
55^    (A)-(t)^:A(f)-*;    VK-2>0:V(«-2>):-^(«-«>). 

56.  |a^62.|5^^3.|(j-i5-i.    (5  +  2V2)-(5-2V2)^ 

57.  (2  +  V3)';    (8V2  +  2V3)-(2V2  +  V3'). 

58.  (4  +  V2)-(l-V3)-(4-V2)-(5-V3)-(l  +  V3)(5  +  V3). 

59.  (a  +  b)^-{a-hby^'{a-b)^'{a-b)n.{a^-^b^)^n; 

^-a-^-b'-^-a-^-b'-^-a'-^-b'^-a'^-b. 

§  9,  PROB.  6. 

•  ••64.   Find  the  required  powers  or  roots  : 

60.  (3V3)^  (2^5)«;  (V2-V3)^  W^O-^oY;  (3*-3-*)2. 

61.  (Vl-Vf)';    (2*-2-t)3;    (3*-3-*)^    (4*+4-*)^ 

62.  (Vl^-|2/)';  [a'&(a'5c)^]*;  (2x^y^z^y',  l{5x^y-^)^Y^, 

63.  ^-2-a'"6"'c2"»;   ■^{27a^x)^',    (a^x-'^  +  cC^xy. 

64.  [(a  +  6)*-(a-6)*]2;   (a*+&*- c*)^  (a*H+a~' 


66. 
67. 
68. 
69. 


§10.]  EXAMPLES.  231 

§  9,    PROB.  7. 

•  ••69.  Reduce  to  equivalent  fractions  with  rational  denominators  ; 

'''    V3'    V^'    2V3'    2V2'   3^^'    5t'   Sy^'    W    '    \n)     ' 
2       .    V2-1.    V3-V2.  21  1  +  V5 

V3  +  1'    V2  +  1'    V3  +  V2'    VIO-V^'   3V5-2V3* 

15 .    (a4-6)^  +  (a-6)^ 

V10  +  V20-f  V^O-V^-V^^'    (a  +  &)^-(a-6)i* 
(3  +  V3)-(3  +  V5)-(V5-2).    1 

(5-V5)-(l  +  V3)  '   a+V[^+V(c  +  V^)] 

V2.(V3  4-l)-(2-V3)  V2^(V2-3) 

(V2-l)-(3V3-5).(2+V2)'    (V2  +  8).(V3-V5)' 

•  ••71.   Reduce  to  simplest  form  : 

r-Q  1 1  .  x+(a^-l)^  .  a;-(a^-i)^ 

a-(a'-a^)i      a+ia'-x')^'  x-{x'-l)^'  x-\-{x'-l)^ 

^^     (a^+l)i4-(fl^-l)^  ^   (x^  +  l)i-.(a^-l)i  ^  x-(x'-\-l)i 
(a,^+l)^_(a;2_i)i      (a,^+i)*  +  (a^_l)r  a;+(a;2-f.l)^ 

72.  In  the  equation 

(a^_  2/6)  :  (^x  —  ?/)  =  ar'^H-  x^y  +  ar^?/^  +  a^?/-^  +  xy^-\-f 

put  a;  =  a^  and  2/  =  6^  ;  thence  find  (a'^  —  Ir)  :  (a-  —  6^) , 
and  apply  this  and  similar  results  to  reduce  to  equivalent 
fractions  with  rational  denominators  : 

1  10         3^-2^.    -^•5--^4.        1         2a+b^ 

a*_6*'   2-^6'  3i+2^'    ^5  +  ^4'  ^3_^i '  g^-^i* 

73.  Show  that 

form  a  complementary  group ;  and  thence  reduce  to 
equivalent  fractions  with  rational  denominators  : 

1  14-V2    .      i-yg    , 

-v/a+V^+V^-'  1  +  V2+V3'  i-V«-V^' 

V2  -  V3      .      a 

1  +  V2  -  V3  '      V(^  +  Vc)  +  V(f^  +  V^)' 


232  POWERS  AND   ROOTS.  [VIII. 

74.  Find  the  value  of: 

a5+(l+»2)^      x-{l^x')^  2|_V6y       \aj  J 
a^ H i^^ — ->  when  x  =  W3. 

75.  Show  that  2/  =  i(e*-e-*)    if  e='  =  2/  + V(l +2/")- 

§  9,  PEOB.  8. 

•  ••78.    Find  the  square  root  of: 

76.  7  +  2V10;    7  +  4V3;    2  -  V3  ;  16-6V7;  V^S-V^^^ 

77.  8V3-6V5;    75-12V21;    V27  +  V15;    -9  +  6V3^ 

78.  aft  +  c^+VCa'-c^X^'-O;   2[H-(l-0^]; 

a;^-2a;(a;2/-ic2)*;  l-2aV(l-a'). 

79.  Find  the  fourth  root  of : 

28-16V3;   49  +  20V6;   a^+b^-^6ab-4.{ah^+ah^). 

80.  If  V^*+V2/  +  V^  =  V(«+2V2>  +  2Vc  +  2V(«), 

show  that  X,  y,  2  must  satisfy  the  four  conditions 
a;  4-  ?/  4-  z  =  a,   xy  =  b,   xz  =  c,   yz^d, 
and  hence  show  that  the  square  root  of 

6  +2  V2  +  2  ^3  +  2  V6  may  be  found. 

81.  Find  the  square  root  of: 

10  +  2 V6  +  2  VIO  +  2 V15  ;     8  +  2 V2+2 V5  +  2 ylO  ; 
15-2V10-2V21  +  2V35;   11  +  2V6+4V3+6V2. 

82.  Show  that  the  square  root  of  10  +2  V6  +  2  ^U  +  2  V21 

cannot  be  expressed  in  the  form  -^/a  -\-  ^h  +  V^* 

83.  Find  the  square  root  of  : 

15  -  2  V3  -  2  V15  +  6  V2  -  2  V6  +  2  V^  -  2  V30. 

§  9,   PROB.  9. 

84.  Find  the  cube  root  of  : 

7  +  5V2;    16  +  8V5;    22  +  lOV';    38+17V5; 
21  V6  -  23  V5 ;    3a -  2a^  +  (1  4-  2a2)  V(l  -  «')• 


ths.  1-3,  §  1.]  GENERAL  rROPEKTIES.  233 

IX.     LOGARITHMS. 

§1.     GENERAL    PROPERTIES. 

The  logarithm  of  a  number  is  the  exponent  of  that  power 
to  which  another  number,  the  6ase,  must  be  raised  to  give  the 
number  first  named.  [I.  §  1 1 

E.g.^  in  the  equation  a""  =  n,  a  is  the  base,  n  the  number  ;  and 
X  tlie  exponent  of  the  power  of  a  and  the  logarithm 
to  base  a  of  the  number  n. 

The  equation    x  =  log^N    expresses  the  relation  last  named. 

The  equation  n  =  log^~^aj  means  that  N  is  the  number,  a*, 
whose  logarithm  to  base  a  is  a; ;  it  is  read,  n  is  the  anti-loga- 
rithm of  X  to  base  a. 

E.g.^   0=logAl    and   A  =  log"^0,  whatever  a  may  be. 

So,  l  =  log22,  2  =  log39,  3=log464,  4=log5625,.-., 
and  2=log2-M,  9=log3-^2,  64=log4-^3,  625  =  log5-H,  .... 

So,  -l=log2j,  -2  =  log3i,  -3=log4^ij,  -4  =  log5^,..., 
and        -l  =  log,2,    -2=log,9,    -3  =  log^^64,    -4=logL625 -.. 

If  the  base  be  well  known  it  may  be  suppressed,  and  these 
two  equations  may  then  be  written   x  =  logN,  n  =  log"^a;. 

If  while  A  is  constant  n  take  in  succession  all  possible  values 
from  0  to  00,  the  corresponding  vahies  of  x  when  taken  together 
constitute  a  system  of  logarithms  to  base  a. 

Operations  upon  or  with  logarithms  are  therefore  operations 
upon  or  with  the  exponents  of  the  powers  of  any  same  base ; 
and  the  principles  established  for  such  powers  apply  directly  to 
logarithms,  with  but  the  change  of  name  noted  above. 

Theor.  1.    The  logarithm  of  unity  to  any  base  is  zero.  [df.  pwr, 

Theor.  2.  The  logarithm  of  any  number  to  itself  as  base  is 
unity.  [df .  pwr. 

Theor.  3.    To  any  positive  base  -j       ^,,      than  unity,  every 

positive  number  has  one  and  but  one  real  logarithm : 

'^  TdecreZin^^  f^^^^^^^  ^/^'^^  '^^'^^^r.  [VIII.  th.  13 


234  LOGARITHMS.  [IX.tlis, 

Note.  If  either  the  base  or  the  number  be  negative,  tliere 
may  or  may  not  be  one  real  logarithm. 

E.g.,    +100  has  the  logarithm  2  to  base  +10  or  "10, 
and  both  +10  and  "10  have  the  logarithm  ^  to  base  +100  ; 

but  "100  has  no  real  logarithms  to  base  +10  or  ~10, 

nor  has  +10  or  ~10  a  real  logarithm  to  base  ~100. 

So,       *1000  has  the  logarithm  3     to  base  =^10, 
and  *10      has  the  logarithm  ^     to  base  ~1000  ; 

but  '''1000  has  no  real  logarithm  to  base  *10, 

and  '^'lO      has  no  real  logarithm  to  base  '''1000. 

In  this  chapter,  and  in  general  where  logarithms  to  the  base  10 
are  used  as  aids  in  numerical  computations,  the  number  as  well  as 
the  base  is  assumed  to  be  positive  unless  the  contrary  be  stated. 

Theor.  4.    If  the  base  he  positive  and\       ^ -..     than  unity,  the 

logarithms  of  all  numbers  greater  than  unity  are  {^    '  , .    \  of  all 

numbers  positive  and  less  than  unity,  \      ^ .  .     '  [VIII.  lem.  th.  1 2 

Theob.  5.  If  the  base  be  positive  and^  ^ ■,.  than  unity,  and 
if  the  number  be  a  positive  variable  that  approaches  zero,  then 
Us  logarithm  approw^Us  {  pZal^^' l^ff^'  [VIII.  th.  13 

Theor.  6.    The  logarithm  of  a  •{  ^      .    .of  two  numbers  is  the 

J  sum  of  the  logarithms  of  the  factors.  rVITT  tl  q  ^  9 

'  excess  of  log.  div'd  over  log.  divW.  -         •       •   ' 

^'9' »    log^  (b  .  c  :  d)  =  log^B  +  log^c  -  log^D. 

Theor.  7.    Tlie  logarithm  of  a-{\  ^^^^  of  a  number  is  the 

^  fuotett  ""^^^'^  logarithm  of  the  number  by  the{  ^ZThxlex. 

[VIII.ths.4, 10 
E.g.,   log^(B2.^c)  =  21og^B+Jlog^c. 

CoR.    The  logarithm  of  the  square  root  of  the  product  of  two 
numbers  is  the  half  sum  of  their  logarithms  to  the  same  base.  [th.  6 
E.g.,  log^VC^-c)  =i(}og^B-\-\ogj,c). 


4-8,  §1.] 


GENERAL  PEOPERTIES. 


235 


Theor.  8.  If  the  logarithm  of  any  same  number  he  taken  to 
two  different  bases,  the  first  logarithm  equals  the  product  of  the 
second  logarithm  into  the  logarithm  of  the  second.base  taken  to  the 
first  base,  and  vice  versa. 

Let  N  be  any  number,  a,  b  two  bases  ; 


then  will  log^  n  =  logs  n  •  log^  b  ,   and    logs  n 


logBN 


log^N-logBA. 

[df.log 


For  let  y 
then   •••  N  =6", 
and  log^N  =  y  •  log^B,  [th.  7 

.-.    log^N  =  logBN.log^B.  Q.E.D. 

So,         l0gBN  =  l0g^N.l0gBA.  Q.E.D. 

Cor.  1 .   log^  b  •  log^  c  •  log^  d  =  log^  d  . 

logj,B'log^C'logoT)  •••  log^i.  =  log^j.. 

Cor.  2.    The  logarithms  of  two  numbers,  each  taken  to  the 
other  number  as  base,  are  reciprocals. 

For  let  N  =  A ; 
then         log^  B  •  logjj  a  =  log^  a  =  1 . 

CoR.  3.    logj^B ' log^C' log^^A  =  1  ; 

log^B'log^C'log^-D  ...  log^A  =  l. 
Note.   The  reader  will  observe  that  th«  bases 
and  numbers  run  in  cyclic  order : 

Cor.  4.    Tlie  modulus  of  any  system  of  logarithms  is  the  loga- 
rithm, in  that  system,  of  the  Napierian  base  e.     [VIII.  th.  15  nt. 
Let  A  be  the  base  of  any  system  of  logarithms,  and  m^  the 
modulus ; 
then   •.•  logAa;  =  logAe.log^a;,  [th. 8 

wherein  log^e  is  a  constant,  independent  of  a;, 
.-.  D,log^a;  =  log^e.Djogea;, 


I.e.,         — ^ 


1 


logA^--. 


X 

.'.  M^  =logj,e. 

E.g.,    M,o  =  logioe  =  logio2.71828 
=  .4342944.... 


[VII.th.16 


Q.E.D. 


236  LOGARITHMS.  [IX.th.9 

§  2.    SPECIAL  PROPERTIES,  BASE  10. 
The  logarithm  of  an  exact  power  of  10  is  an  integer,    [df.log 
E.g.,    of..-,  1000,  100,  10,  1,   .1,  .01,  .001,..- 
the  logarithms  to  base  10  are 
...,      +3,     +2,  +1,  0,  -1,    -2,       -3,  .... 
But  of  any  other  number  the  logarithm  is  fractional  or  incom- 
mensurable, and  consists  of  a  whole  number,  the  characteristic.! 
and  an  endless  decimal,  the  mantissa.  [VIII.  §  4  df.  incom.  pwr. 
As  a  matter  of  convenience  the  mantissa  is  always  taken  posi- 
tive ;  and  the  characteristic  is  the  exponent,  positive  or  negative, 
of  the  integral  power  of  10  next  below  the  given  number. 
A  negative  characteristic  is  indicated  by  the  sign  —  above  it. 
E.g.^  of  the  numbers 

2000,  20,  .2,  .002, 

the  logarithms  to  base  10  are 

3.30103...,  1.30103...,  1.30103...,  3.30103.-., 
whose  characteristics  are   3,    1,     1,    3, 
and  whose  common  mantissa  is  +.30103 .... 
Theor.  9.    If  a  number  be  multiplied  (or  divided)  by  any 
integral  power  o/ 10,  the  logarithm  of  the  product  (or  quotient) 
and  the  logarithm  of  the  number  have  the  same  mantissa. 
For     •.•  the  logarithm  of  a  product  is  the  sum  of  the  logarithms 
of  its  factors.  [th.  6 

and    •.•  the  logarithm  of  the  multiplier  is  integral,  [hyp. 

.-.  the  mantissa  of  the  sum  is  identical  with  the  mantissa 
of  the  logarithm  of  the  multiplicand.         q.e.d. 
So,       if  a  number  be  divided  by  an  integral  power  of  10. 
CoR.    For  all  numbers  that  consist  of  the  same  significant  fig- 
ures in  the  same  order,  the  mantissa  of  the  logarithm  is  constant, 
but  the  characteristic  changes  with  the  position  of  the  decimal  point 
in  the  number. 

E.g.,  of  the  numbers 

79500,    795,        7.95,       .0795,     .000795, 
the  logarithms  to  base  10  are 

4.9004,  2.9004,  0.9004,  2.9004,  4.9004. 


pr.  1,  §  3.]  COMPUTATION  OF  LOGARITHMS.  237 

§  3.     COMPUTATION  OF  LOGAEITHMS. 

PrOB.  1.  To  COMPUTE  THE  LOGARITHM  OF  A  NUMBER  TO  A 
GIVEN  BASE. 

FIRST   METHOD,   BY   CONTINUED   FKACTIONS. 

Form  the  eopponential  equation^  a^  =  n,  wherein  n  is  the 
number^  a  the  base,  and  x  the  logarithm  sought.  [df .  log 

By  trial  find  two  consecutive  integers,  x'  and  x'  +  1 ,  between 
which  X  lies,  and  lorite  x  =  x'  -f  y~^,  wherein  x'  is  known  and 
y~^  is  some  positive  number  less  than  unity. 

In  the  equation  a''  =  n,  replace  :s.  by  x' +  j'^ ,  giving  A'''+y"^=N, 
and  divide  both  members  by  a^',  giving  aj=  n  :  a^',  =  n',  say. 

Raise  both  members  of  the  equation  aj  =  n'  to  the  yth  power, 
giving  a  =  n'^. 

By  trial  find  two  consecutive  integers,  y'  and  y'  +  1,  between 
which  y  lies,  write  y  =  y'  -f-  z~^,  and  so  on,  as  above. 

Then  x  =  x'  +  i  =  x'+^  1  =x'-f  —  i 

y  y'_f_±  y'4-  — 

and  the  convergents,  which  approach  x  as  their  limit,  are : 
,     x'y'  +  \     x'y'z'-{-z'-\-x.' 
'        y'     '      ^y'z'  +  l       '"•* 
E.g.,  given  10*  =  5,  to  find  x,  i.e.  to  find  logio5. 
Put      X  =0-f2/~S 

then   •.•  IQy        =5,  5''  =  10,  2/=  1  + - 

z 

5'"^^      =10,      5'  =  2,  2'=5,   z  =  2  +  - 

2-J      =5,        2-J  =  |,  (|J  =  2,   .  =  3  +  i 

\ij  \ij       125'   Vl25^      4'  ^t 

and  so  on. 

1  +  -     1+^1_1     1+^1  1  +  ^1 

z  2+-  2+—-   1  2-j-—   1 

and  the  convergents  are  ^  9  + 

J     2    _7_    65 
'   3'    10'   93'  "** 


238  LOGARITHJMS.  [IX.  pr. 

These  convergents  are  alternately  too  large  and  too  small ; 

but  their  errors  are  respectively  less  than 

1.  __1_^J^.        1^1.   1 

3'   3-10    30'   10-93      930'   93 •  next  denominator' 

which  denominator  is  not  less  than  93  +  10,  =103  ;  [VI.ths.1,2 

65 
.*.  — ,    =. 69892 •••,   is  too  small,  and  differs  from  the 
93  1 

tiue  value  by  less  than 

^  9579 

The  true  logarithm  of  five  to  seven  decimal  places,  as  shown 

by  the  tables,  is  .6989700,  so  that  —  actually  differs  from  it  by 

93 

less  than  half  of  one  ten-thousandth. 

So,  logio2  =logiolO  -  logio5  =  1  -  .69897  =  .30103. 
So,  log 4     =2.1og2=   .60206  ;log8=3.1og2        =   .90309; 
log625=4.1og5  =  2.79588 ;  Iog4=log4-log5  =  1.90309. 

SECOND   METHOD,   BY   SUCCESSIVE   SQDAKB    ROOTS   OF   PRODUCTS. 

Take  two  numbers  wJiose  logarithms  are  known ^  the  one  greater 
and  the  other  less  than  the  given  number. 

Find  the  square  root  of  their  product  and  the  logarithm  of  this 
root^  the  half  sum  of  their  logarithms.  [th.  10 

Multiply  this  root  by  whichever  of  the  two  numbers  lies  at  the 
other  side  of  the  given  number^  and  find  the  square  root  of  the  prod- 
uct, and  the  half  sum  of  the  logarithms  of  the  factors ;  and  so  on. 

E.g.,  to  find  the  logarithm  of  5  to  the  base  10 : 

Take  10  whose  logarithm  is  1,  and  1  whose  logarithm  is  0 ; 

Number. 
then  V{10X1)  =3.16227766; 

V(10X  3.16227766)  =5.62341325; 
V(3.16228  X  5.62341)  =  4.21696535 ; 
V(5.62341  X  4.21697)  =  4.86967671 ; 
V{5.62341  X  4.86968)  =  5.23299218 ; 
V(4.86968  X  5.23299)  =  5.04806762 ; 
V(4.86968  X  5.04807)  =  4.95807276 ;  |(.6875  +  .703125)  =  .69531 
V(4.95807  X  5.04807)  =  5.00028680 ;  |(. 69531  +  .70312)  =  .69921 
V(4.95807  X  6.00029)  =  4.97709632 ;    I  (.69921  +  .69531)  =  .69726 

|(.69921  +  .69726)  =  .69823 


Logarithm. 

^(1  +  0) 

=  .5 

A(l  +  .5) 

=  .75 

K.5  +  .75) 

=  .625 

i(.75  +  .625) 

=  .6875 

^(.75 +  .6875) 

=  .71875 

K.6875  +  . 71875) 

=  .70312 

1,§4.]  TABLES  OF  LOGARITHMS.  239 

§  4.   TABLES  OF  LOGARITHMS. 

If  for  successive  equidistant  values  of  a  variable  the  corre- 
sponding values  of  a  function  of  this  variable  be  arranged  in 
order,  the  function  is  tabulated;  the  variable  is  the  argument  of 
the  table  [I.  §  13]  and  the  successive  values  of  the  function  are 
the  tabular  numbers.  The  values  of  the  argument  are  commonly 
placed  in  the  margin  of  the  table. 

If  the  logarithms,  to  any  one  base,  of  the  successive  integers 
from  1  to  a  given  number,  say  1000,  or  10000,  be  arranged  for 
ready  reference,  they  form  a  table  of  logarithms.  Such  tables 
are  in  use  to  three  places  of  decimals,  to  four,  five,  six,  seven, 
and  even  ten,  twenty,  or  more  places. 

In  general,  the  greater  the  number  of  decimal  places,  the 
greater  the  accuracy,  and  the  greater  the  labor  of  using  the 
tables.  For  the  ordinary  use  of  the  engineer,  navigator,  chem- 
ist, or  actuary,  four-  or  five-place  tables  are  sufficient ;  but  most 
refined  computations  in  Astronomy  or  Geodesy  require  at  least 
seven- place  tables. 

Most  logarithmic  tables  are  arranged  on  the  same  general 
plan  as  the  four-place  table  given  on  pp.  248,  249.  This  table 
gives  the  mantissa  only  ;  the  computer  can  readily  supply  the 
characteristic.  To  save  space,  the  first  two  figures  of  each 
argument  are  printed  at  the  left  of  the  page,  and  the  third  figure 
at  the  top  of  the  page  over  the  corresponding  logarithm. 

To  save  time,  labor,  and  injury  to  the  eyes,  the  computer 
should  use  a  well-arranged  table,  and  then  train  himself  to  cer- 
tain hal)its.  The  best  tables  have  the  numbers  grouped  by 
spaces,  or  by  light  and  heavy  lines,  into  blocks  of  three  or  five 
lines,  and  three  or  five  columns,  corresponding  to  the  right-hand 
figures  of  the  arguments  of  the  table.     The  usual  patterns  are 

|0|1  2  3|4  5  6|7  8  9|0|  1  2  3|  ...for  three-line  blocks, 
and       |0  1  2  3  4|5  6  7  8  9|0  1  2  3  4| ...  for  five-line  blocks, 
as  in  the  table  on  pp.  248,  249.     Instead  of  tracing  single  lines 
of  figures  across  the  page  and  down  the  column,  the  computer 
should  learn  to  guide  himself  by  correspondences  of  position  in 
the  blocks. 


240  LOGARITHMS.  [IX.  pr. 

§5.     OPERATIONS  WITH   COMMON   LOGARITHMS. 

PrOB.   2.     To   TAKE    OUT   THE    LOGARITHM    OF  A  GIVEN  NUMBER. 

(a)    One,  two,  or  three  significant  fixtures. 

If  the  number  have  one  significant  figure,  annex  two  zeros; 
if  two  significant  figures,  annex  one  zero;  for  the  mantissa 
write  the  four  figures  that  lie  opposite  the  first  two  figures  and 
under  the  third  figure,  and  for  the  characteristic  write  the  exponent 
of  the  power  o/ 10  next  below  the  given  number. 

E.g.,    log  567  =  2.7530;  log  5.6  =  0.7482  ;  log  .05  =  2.6990; 

If  a  number  have  more  than  three  significant  figures,  the 
mantissa  of  its  logarithm  is  not  found  in  the  table,  but  lies 
between  two  tabular  mantissas  whose  arguments  are  two  three- 
figure  numbers  next  larger  and  next  smaller  than  the  given 
number.  [th.  3 

E.g.,    mantissa  log 500.6  lies  between  .6990,  .6998, 
i.e.,  between  mantissa  logs  500,  501. 

(6)  Four  or  more  significant  figures. 

Find  the  mantissa  of  the  logarithm  of  the  first  three  figures  as 
above;  subtract  this  mantissa  from  the  next  larger  tabular  man- 
tissa, and  take  such  part  of  the  difference  as  the  remaining  figures 
are  of  a  unit  having  the  rank  of  the  third  figure;  add  this  2')rod- 
uct,  as  a  correction,  to  the  mantissa  of  the  first  three  figures. 

E.g.,  to  find  log  500.6; 
then   •.•  log 500  =  2.6990,    log 501  =  2.6998,  [tables 

and  log  501 -log  500  =.0008,    500.6 -500  =  .6, 

.-.  log 500. 6  =  2. 6990 +  .6  of  .0008=2.6995. 

Note  1 .  If  the  given  number  lie  nearer  the  larger  of  the  two 
arguments,  its  mantissa  is  easiest  found  by  subtracting  from 
the  larger  of  the  two  tabular  mantissas  such  part  of  their  dif- 
ference as  the  excess  of  the  larger  argument  over  the  given 
number  is  of  a  unit  having  the  rank  of  the  third  figure. 

E.g.,  to  find  mantissa  log  500.6  ; 
then   •.•  mantissa  logs  500,  501  =  .6990,  .6998,  [tables 

and     •••  log 501— log 500  =  .0008,    501 —500.6  =  .4, 

.-.  mantissa  log 500.6  =  .6998  —  .4  of  .0008  =  .6995. 


2,  §5.] 


OPERATIONS    WITH    COMMON  LOGARITHMS. 


241 


or 


Note  2.  The  rule  for  interpolating  or  applying  the  correction 
rests  upon  a  property  which  logarithms  have  in  common  with 
most  other  functions,  and  which  the  reader  may  observe  for 
himself  if  he  will  examine  the  table  carefully,  viz.  :  that  the 
differences  of  logaritlims  are  very  nearly  proportional  to  the  dif- 
ferences of  their  numbers  when  these  differences  are  small.  They 
are  not  exactly  proportional,  but  the  error  is  so  small  as  to  be 
inappreciable  when  using  a  four-place  table.  The  seven-place 
tables  give  the  logarithms  of  all  five-figure  numbers,  and  the 
errors  for  the  sixth,  seventh,  and  eighth  figures,  as  far  as  due  to 
this  cause,  are  inappreciable.  So  the  rule  above  given  "for  ap- 
plying the  correction  "  is  universal. 

Note  3.  The  computer  should  train  himself  to  find  the  correc- 
tion and  add  it  to  the  tabular  mantissa  (or  subtract  it)  mentally, 
and  to  write   down   only   the  j^q       jg 

final  result. 

To  aid  in  this  mental 
computation,  small  tables  of 
proportional  parts  are  often 
printed  at  the  side  of  the 
principal  table.  Two  forms  of 
such  tablets  are  here  shown  : 
the  first  most  accurate,  and 
the  other  of  easiest  use. 

E.g.^  to  find  mantissa  log  22674  ; 
then    •.•  log  227 -log  226  =  .3560 -.3541  =  .0019, 
.*.  the  correction  to  be  added  to  .3541  is 

.7  of  .0019  +  .04  of  .0019  ;  and  is  found  thus  : 
opposite  7  find       13.3     or     13  .3541 

opposite  4  find       ^  __1  +14 

Add  ;   the  correction  is     14  14  giving  .3555 

Or  •.•  22700-22674  =  26, 

.'.  the  correction  to  be  subtracted  from  .3560  is 

.2  of  .0019  +  .06  of  .0019  ;  and  is  found  thus  : 
opposite  2  find         3.8     or       4  .3560 

opposite  6  find         1.1  1  —5 

Add ;  tlie  correction  is       5  5  .3555 


1.9 

1.8 

3.8 

3.6 

5.7 

5.4 

7.6 

7.2 

9.5 

9.0 

11.4 

10.8 

13.3 

12.6 

15.2 

14.4 

17.1 

16.2 

19  18 

1 

2 

2 

2 

4 

4 

3 

6 

5 

4 

8 

7 

5 

10 

9 

6 

11 

11 

7 

13 

13 

8 

15 

14 

9 

17 

16 

242  LOGARITHMS.  [IX.  prs. 

PrOB.  3.      To   FIND   A   NUMBER   FROM   ITS    LOGARITHM. 

(a)    The  mantissa  found  in  the  table. 

Write  down  the  two  figures  opposite  to  the  given  mantissa  in 
the  lefi-hand  column^  and  following  them  the  figure  at  the  top  of 
the  column  in  which  the  mantissa  is  found. 

Place  the  decimcUpoint  so  that  the  number  shall  he  next  above 
that  power  0/ 10  ichose  exponent  is  the  given  characteristic. 

JE;.gr.,log-i2. 7536=567  ;log-^0.7482  =  5.6;  log"^  2.G990  =  .05. 

(6)   The  mantissa  not  found  in  the  table. 

Take  out  the  first  three  figures  for  the  tabular  mantissa  next 
less,  as  above;  from  the  given  mantissa  subtract  this  tabular 
mantissa,  and  divide  the  difference  by  the  difference  between  the 
tabular  mantissa  next  less  and  thai  next  greater. 

Annex  the  quotient  to  tlie  three  figures  first  found. 

Place  the  decimal  point  as  above. 

E.g.,  to  find  log"^  2.6995. 
then   •.•  log-i  2.6990  =  500,    log-^  2.6998  =  501,  [tables 

and     •.•  2.6995— 2.6990  =  .0005,     2.6998  —  2.6990  =  .0008  ; 
.-.  the  number  sought  is  500  +  (.0005  :  .0008) ,  =  500.6. 

Note  1.  The  process  is  but  the  inverse  of  that  for  taking 
out  logarithms,  and  the  reason  of  the  rule  is  the  same  for  both. 

This  four-place  table  allows  only  one-figure  corrections,  and 
so  gives  only  four- figure  numbers.  In  general,  an  n-place  table 
gives  7i-figure  numbers ;  but  sometimes,  when  the  mantissa  is 
large,  the  ?ith  figure  may  be  two  or  three  units  in  error,  and 
then  the  number  is  approximate  only  for  n—1  figures  [V.  §5]. 

Note  2.  If  the  given  mantissa  lie  nearer  the  larger  of  the  two 
tabular  mantissas,  the  correction  may  be  applied  to  the  larger 
argument  by  subtraction. 

E.g.,  to  find  log"^  .3555  ; 
then   •.•  the  next  tabular  mantissas  .3541,  .3560  differ  by  .0019, 

and  correspond  to  226,  227,  as  arguments, 
and     •.•  .3555 -.3541  =  .0014,    .3560  — .3555  =  .0005, 

.-.  the  number  sought  is  226  -f  \^,  or  227  —  3%'  =  22G74. 


3-5,  §  5.]    OPERATIONS  WITH  COMMON  LOGARITHMS.  243 

If  the  tablets  of  proportional  parts  be  used,  the  work,  writteu 
out,  appears  as  follows  : 

14  226  or             5  227 

13.3  +.7                         3^               -.2 

.7  +    4                       1.2                -    6 

.8  226.74                       1.1  226.74 

PrOB.  4.  To  FIND,  BY  ONE  OPERATION,  THE  ALGEBRAIC  SUM 
OF  SEVERAL  LOGARITHMS. 

Arrange  the  logarithms  vertically,  and  take  the  algebraic  sum 
of  ea/ih  column  of  digits,  beginning  at  the  right  and  carrying  as 
in  ordinary  addition;  if  this  sum  for  any  column  be  negative, 
make  it  positive  by  adding  one  or  more  tens  to  it  and  subtract  as 
many  units  from  the  next  column. 

E.g.,   to  find  the  algebraic  sum  in  the  margin,       3.1037 
adding  upward,  the  computer  says :      —  0.6986 

9,    7,  16,    10,    17,  +2.2409 

1,    3,-6,-14,-11,    9,  2  off,         -2.5892 

-2,    3,-5,    -1,-10,    0,  1  off,         +1.2529 

-1,    1,-4,    -2,    -8,-7,   3,     1  off,         =1.3097 
-1,-2,    0,    -2,      1, 
and,         adding  downward,  for  a  check,  he  says : 

7,    1,  10,      8,     17, 

1,    4,-4,-13,-11,    9,  2  off,     and  so  on. 

PrOB.  5.  To  DIVIDE  A  LOGARITHM  WHOSE  CHARACTERISTIC  IS 
NEGATIVE. 

Write  down  the  number  of  times  the  divisor  goes  into  that  mul- 
tiple of  itself  which  is  equal  to,  or  next  less  than,  the  negative 
characteristic  ;  carry  on  the  positive  remainder  to  the  mantissa, 
and  divide. 

E.g.,    4.1234  :  3  =  ("6  +  2.1234)  :  3  =  2.7078. 

So,       3.4770.  I  =     8.4310:2  =4.2155. 


244  LOGARITHMS.  [IX.  prs. 

PrOB.  6.     To    AVOID   NEGATIVE    CHARACTERISTICS. 

Modify  the  logarithms  by  adding  10  to  their  characteristics  when 
negative;  use  the  sums,  differences,  or  exa^  multiples  of  the 
modified  logarithms  where  the  subject-matter  is  such  that  the  com- 
puter cannot  mistake  the  general  magnitude  of  the  results. 

To  divide  a  modified  logarithm,  add  such  a  multiple  of  10  as 
will  make  the  modified  logarithm  exceed  the  true  logarithm  by  10 
times  the  divisor;  then  divide. 

E.g.,  if  loga  =  2.3010,  log6=1.4771,  to  find  log (a^ 6~^) , 
=  i(2\oga-3logb). 

BY  TRTTE  LOOABITHMS. 

2.3010.2  =  4.6020 

1.4771.3  =  2.4313 


BY  MODIFIED  LOGARITHMS. 

8.3010-2  =  6.6020 
9.4771.3  =  8.4313 


5)2.1707  5)8.1707 

T.6341  9.6341 

At  each  step  of  the  work  with  modified  logarithms,  any  tens 
in  the  characteristics  are  rejected,  or  tens,  if  necessary,  are 
added,  so  as  to  keep  the  characteristics  between  0  and  9  inclu- 
sive. Before  dividing  by  5,  in  the  example  just  above,  4  tens 
were  added,  making  the  dividend  48.1707. 

Note.  The  arithmetical  complement  of  the  logarithm  of  a 
number  is  the  modified  logarithm  of  the  reciprocal  of  the  num- 
ber. It  is  got  by  subtracting  the  given  logarithm,  modified,  if 
necessary,  from  10  ;  it  may  be  read  from  the  table  by  subtract- 
ing each  figure  from  9,  beginning  with  the  characteristic  and 
ending  with  the  last  significant  figure  but  one,  subtracting  the 
last  significant  figure  from  10,  and  annexing  as  many  zeros  as 
the  given  logarithm  ends  with.  The  arithmetical  complement 
of  the  arithmetical  complement  is  the  original  logarithm. 

E.g.,    ar-com 3.4908000  =  6.5092000,    and  conversely. 

In  any  algebraic  sum,  a  subtractive  logarithm  can  be  replaced 
by  its  arithmetical  complement  taken  additively.  In  most  cases, 
however,  the  method  of  prob.  4  appears  preferable. 

E.g.,  in  the  example  under  prob.  4,  the   terms    —0.6986. 
-2.5892  might  be  replaced  by  9.3014,  1.4108. 


6-8,  §  5.]     OPERATIONS  WITH  COMMON  LOGARITHMS.  245 

PROB.  7.  To  COafPUTE  BY  LOGARITHMS  THE  PRODUCTS,  QUO- 
TIENTS,  POWERS,   AND   ROOTS    OF   NUMBERS. 

1 .  For  a  product :  add  the  logarithms  of  the  factors^  and  take 
out  the  antilogarithm  of  the  sum. 

2.  For  a  quotient:  from  the  logarithm  of  the  dividend  subtract 
that  of  the  divisor^  and  take  out  the  antilogarithm. 

3.  For  a  power :  multiply  the  logarithm  of  the  base  by  the  ex- 
ponent of  the  power  sought^  and  take  out  the  antilogarithm, 

4.  For  a  root:  divide  the  logarithm  of  the  base  by  the  root- 
index^  and  take  out  the  antilogarithm. 

E.g.,  to  find  the  value  of  (.01519.6.318:7.254)': 

KVMBEBS.  LOGABITHMS. 

.01519  2.1815 

X  6.318  -f  0.8006 

H- 7.254  -0.8605 

2.1216x1 
and  the  number  sought  is  0.001522.  3.1824 

Note.  Not  only  simple  operations,  as  in  the  above  example, 
but  complex  operations,  can  be  performed  by  logarithms.  Some- 
times the  expression  whose  value  is  sought  must  first  be  prepared 
by  factoring. 

E.g.,  to  find  the  value  of  ^{h^—b^),  wherein  7i,  6  are  any 
given  numbers  and  may  represent  the  lengths  of  the 
hypothenuse  and  base  of  a  right  triangle  : 


then         V(^'-  ^')  =  log-H(log/i  +  6  +  logh  -  b). 

Prob.  8.    To  SOLVE  the  exponential  equation   a*  =  b. 

Divide  the  logarithm  of  b  by  the  logarithm  of  the  base  a  of  the 
exponential:  the  quotient  is  x,  the  exponent  sought. 

For     •.•  A='  =  B, 

.*.    CCl0gA  =  l0gB, 

.-.  a;  =  logB  :  logA.  q.e.d. 


246  LOGAEITIOIS.  [IX.  pr. 

FrOB.  9.  To  ESTIMATE  THE  AMOUNT  OF  POSSIBLE  ERROR  IN  A 
LOGARITHM  OR  ANTILOGARITHM  GOT  FROM  THE  TABLE,  AND  IN 
THE   SOLUTIONS   OF  PROBS.    7,    8  : 

Let  p  he  the  number  of  decimal  places  in  the  table  used; 
a',  b',  •••  x',  (a" b'' •••)',  the  number  of  units  of  their  last  decimal 
places  contained  m  a,  b,  •  •  •  x,  a" b°  •  •  • ;  a,  ^,  •  •  •,  ^^e  possible  rela- 
tive errors,  all  taken  positive,  o/a,  b,  ••• :  then 
(a)  Poss.  err.  log  x  =  10~p  +  .43  poss.  rel.  err.  x. 
{b)  Poss.  rel.  err.  x  =  1 :  2x'H-  2.3  -poss.  err.  log  x. 
(c)   Poss.  rel.  err.  a""  b"*  •  •  •   \_in  pr.  7] 

=  1:2  (a'^b".-.)'  -f-  2.3  (+m  -{-+n  +  •.•)  •  IQ-p 
H-(+ma ++0^8 +  ...). 
{d)  Poss.  rel.  err.  x  [in  pr.  8] 

^    1        10-P+.43a      10-P+.43;3 
2x'  log  A.  logB 

For    *.*  DjlogioX  =  Mio -7  [Vni.  th.  15,  A  =  10 

X 

.  • .  Mio  =  X  •  Dj  logio  X  ==  — i—  =  .43 ,  [table  logs 

incx 

inc  log  X  ^  .43 
incx     ~  X 

I.e.,  inclogx=.43 , 

-  inc  X        .  inc  log  x      ^  _  .       , 

and  = -P— =  2.3  inc.  log x: 

x  .4d 

(a)     •.*  logx,  as  got  from  x  by  p-place  logarithm-tables,  has  a 

.   possible  error  composed  of  : 

two  possible  half -units  in  pth  decimal  place,  from  the 

omitted  decimals  of  the  printed  logarithm  and  of  the 

correction  for  interpolation, 

,                .               ^                   .     .„   inc.  or  err.  of  X      ^  . 
and  an  increment  or  error,  =  .43 ;    [above 


poss.  err.  logx  =  (^  +  ^)10-''  +  .43 


X 

err.  X 


X 

=  10~^-f  .43-poss.rel.err.x.     q.e.d. 

(6)     •.•  X,  as  got  from  logx  by  the  same  table,  has  a  possible 
error  composed  of : 


0,  §  5.]      OPERATIONS   WITH  COINBION  LOGARITHMS.  247 

a   possible   half-unit  in   last   decimal   place,  for  the 
omitted  decimals, 
and  an  increment  or  error,  =  2.3  •  x  •  inc.  log  x, 

. • .  poss.  eiT. X  =  |-  in  last  decimal  place  of  x-f  2.3  •  x  •  poss. 

err.  logx, 
.-.  poss.  rel.  err.  X  =  1 :  2x' 4-2.3' poss.  err.  logx.   q.e.d. 

(c)  •.•  poss.  err.  log  A  =  10~^-f  .43a,  [(a) 
.  • .  poss.  err.  log  a"*  =  m  (lO"*  +  .43  a 

So,       poss.err.  logB"  =n(10-^-f  .43/5),  .-•, 
.*.  poss.  err.  log  (a'"b'*"-) 

=  (+??i  -h+w  +...)  10-^  +  .43(+ma  ++np  +  •••)' 
.*.  poss.  rel.  err.  (a'"b**---) 

=  1  :  2x'4-2.3[(+m++n  +  -.-)10-^-f.43(+ma-f  ..•)], 
wherein   x  =  a"*  b"  •  •  •  ; 

.*.  poss.  rel.  err.  (a'"b"-") 

=  1  :2x'-f-2.3(+m4- +w -}-•••)  lO^^'+C"^ '"*«  +  •••)  • 

Q.E.D. 

(d)  •.'  x  =  logB:logA, 

.'.  poss.  rel.  err.  x= poss.  rel.  err.  from  omitted  decimals  of  x 
-fposs.  rel.  err.  log  a + poss.  rel.  err.  logB  [V.  th.  5  cr.  3 

2X'^  log  A  ^  lOgB.  ^  ^^    ^ 

Note.   If  in  (d)  the  divisions  log  b  :  log  a  be  performed  by 
logarithms, 
then   • .  •  log  X  =  log  log  b  —  log  log  a, 

.*.  poss.  err. logx=poss.  err. log-log A-f-poss.  err. log-log b 
=  10~^  +  .43  poss.  rel.  err.  log  a 
-f  10^  +  .43  poss.  rel.  err.  logB  [(a) 


=  2 .  10- +  .43  ri5Z±i43a  _^  10-+,43^\ 

V         log  A  logB         J 

+  2.3  poss.  err.  logx 
10-^ +  .43 a  ,  10-^ +.43^ 


poss.  rel.  err.  x,  =  — ^  +  2.3  poss.  err.  logx  [(6) 

ji  X 


=  -^+4.6-10-^  . 

2  X'  loo^  A  log  B 

which  differs  from  the  former  result  only  by  the  term  4.6-10"'' 
arising  from  the  omitted  decimals  of  the  table  used 
in  performing  the  division,  and  obtainable  also  from 
(c)  by  making  m  =  n  =  l,  a  =  ^  =  0. 


248 


LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

OiU 

0453 

0492 

0531 

0569 

0607 

0645 

0082 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1100 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1307 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1014 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2270 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2705 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3706 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

6105 

5119 

6132 

5145 

6159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

6250 

6263 

5276 

6289 

6302 

34 

5315 

5328 

5340 

5353 

6366 

6378 

6391 

5403 

5416 

6428 

35 

5441 

5453 

5465 

5478 

5490 

6502 

6514 

6527 

6539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

6635 

5647 

5058 

5670 

37 

56S2 

5694 

5705 

5717 

6729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

6843 

5855 

6866 

5877 

6888 

5899 

39 

5911 

5922 

5933 

5944 

6955 

5966 

6977 

6988 

5999 

6010 

40 

■602i 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

0212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

0415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6605 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7120 

7135 

7143 

7152 

52 

7100 

7168 

7177 

7185 

7193 

.,  7202 

7210 

7218 

7220 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

LOGARITHMS. 


249 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7G34 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

80G2 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

^62 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8GSG 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9+84 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

98S6 

9890 

9894 

9899 

9903 

9908 

»8 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991. 

9996'. 

250  LOGAEITHMS.  '  [IX.  §G. 

§  6.     EXAMPLES. 

1.  What  is  the  logarithm  of  144  : 

to  base  2  V3?   to  base  2^12?   to  base  (2^12) -i? 

2.  What  is  the  characteristic  of : 

logoT?  log72?   log321?   logoiS?   log^21?   log,21-i? 

3.  Find  log53125  ;   log7343-M    log  1 81;   log,  343;  log,343-\ 

§  3,   PEOB.  1. 

4.  By  continued  fractions  derive  the  logarithms,  to,  base  10,  of 

3  and  7  to  four  decimal  places. 
Thence  find  the  logarithms  of : 

9,  2.7,  .81,  70,  4.9,  343,  21,  63,  .441,  .7-S    18.9'^ 

§  5,    TROB.  2. 

5.  From  the  table  take  out  the  logarithms  of : 

12,  120,  123,  124,  123.4,  1.234,  12350,  .001235. 

§  6,   PROB.  3. 

6.  From  the  table  find  the  antilogarithms  of  : 

1.0792,  2.0792,  2.0899,  2.0934,  2.0913,  0.0913,  4.0917. 

§  5,  PROBS.  4-8. 

7.  By  logarithms  find  the  values  of  : 

2^5^85^     V(97^-9^)     V12^65      -«/83.64  x  39.56^ 
3273   '      81.^572    '    y5--^.i8'    .081452x</1.968' 

8.  From  the  logarithm  of  2  find  the  number  of  digits  in  : 

2^   2^,   51™,   208,   iQQio^   2525,    6.25«^   25-S   50  «>. 

9.  By  logarithms  multiply  575.25  by  l.OG^O;  by  1.03^";  by  1.015«'. 

10.  By  logarithms  find  ^1000,  -^.00010098,  ^.0000000037591. 

11.  Whatpower  is  2of  1.05?  3  of  1.04?  4  of  1.03?  5  of  1.02? 

12.  If  the  number  of  births  per  year  be  1  in  45,  and  of  deaths 

1  in  60,  in  how  many  years  will  the  population  double, 
taking  no  account  of  other  sources  of  increase  or 
decrease  ? 

§  5,  PROB,  9. 

13.  Find  the  possible  error  in  each  of  the  examples  inNos.  7-12. 


X.  §  1.]  DEFINITIONS  AND  GRAPHIC  REPRESENTATION.      251 


X.     IMAGINARIES. 

Before  taking  up  this  chapter,  the  reader  may  refer  to 
what  is  said  of  numbers  in  I.  §  1  and  of  negatives  in  I.  §  3  ; 
and  particularly  the  note  at  the  end  of  I.  §  3.  He  will  observe 
that,  for  some  kinds  of  quantity,  negatives  as  well  as  fractions 
are  impossible.  He  may  not  be  surprised,  therefore,  to  learn 
that,  even  if  the  operation  denoted  by  imaginary  numbers  can 
be  conveniently  performed  upon  only  one  kind  of  magnitude, 
they  have  most  of  the  properties  of  real  numbers  and  play  an 
important  part  in  algebra.  These  operations  can,  however,  be 
performed,  though  less  simply,  upon  all  kinds  of  magnitude, 
as  appears  in  chapter  XV. 

§  1.    DEFINITIONS  AND  GRAPHIC  REPRESENTATION. 

In  measuring  any  thing  some  unit  of  like  kind  is  first  assumed, 
and  the  relation  the  thing  measured  bears  to  this  unit,  both  as 
to  magnitude  and  as  to  sense  or  quality,  is  expressed  by  a  num- 
ber [I.  §  1].  Conversely,  this  number  expresses  that  operation 
which  must  be  performed  upon  the  unit  to  produce  the  thing : 
the  unit  being  then  the  operand,  the  number  the  operator,  and 
the  thing  the  result  of  the* operation. 

POSITIVE   AND   NEGATIVE    NUMBERS. 

In  the  method  of  graphic  representation  of  numbers  here  de- 
scribed, a  finite  straight  line  pointing  in  an  assumed  direction 
is  chosen  as  the  concrete  unit ;  and  the  relation  that  any  straight 

line  pointing  in  the  {  ..     direction  has  to  this  unit  is  ex- 

pressed by  a  ^  P    ^  )7^   number. 
^  "^       '  negative 

If  the  reader  so  place  himself  before  the  unit  that  to  him  it 

becomes  horizontal  and  points  to  the  right, > ,  then  any 

horizontal  line   pointing   to  the  right,   • >,   has  its 

length  and  direction  in  terms  of  the  unit  line  expressed  by  a 
positive  number.  If  the  line  be  taken  up  and  reversed,  so  that 
it  is  still  horizontal  but  points  to  the  left,  -^ ,  then 


252  IMAGINAEIES.  [X. 

the  relation  of  its  length  and  direction  to  the  unit  is  expressed 
by  a  negative  number.  The  length  remains  as  before  ;  but  the 
quality,  or  direction,  is  reversed. 

VECTORS. 

A  vector,  or  directed  right  line,  is  any  line  whose  length  and 
direction  are  considered,  but  not  its  location.  Its  two  extremi- 
ties are  distinguished  from  each  other  as  its  initial  point  and  its 
temiinal  point.  Its  direction  is  the  direction  of  the  terminal 
point  from  the  initial  point,  and  would  be  reversed  if  these 
poihts  were  interchanged. 

As  the  name  implies,  a  vector  may  be  regarded  as  the  repre- 
sentative of  the  operation  of  carrying  a  particle  from  its  initial 
point  to  its  terminal  point. 

The  direction  of  a  vector  may  be  designated  by  the  order  in 
which  its  two  extremities  are  named,  or  by  an  arrow-head. 

Onno^'tp  '^^^^^^  ^^®  those  having  the  same  length  and 
.  the  same  direction. 
'  opposite  directions. 

E.g.,  the  vectors  ab,  cd,  e  are  equal  to  each 
other,  but  are  opposite  to  the  vectors  ba,  dc,/. 

A  vector  quantity  is  any  concrete  quantity 
whose  magnitude  and  direction  only  are  con- 
sidered, and  which  is  naturally  represented  by 
a  measured  and  directed  right  line  or  vector. 

E.g.,  the  direction  and  velocit}'  or  force  of  the  wind,  or  of  an 
electric  current,  is  a  vector  quantity,  and  may  be  represented 
by  an  arrow. 

COMPLETE    REVERSALS. 

When  the  operand  is  a  vector,  the  operation  of  multiplying  it 
by  ~1  consists  in  reversing  its  quality  or  direction,  and  is  ex- 
hibited thus : 

multiplier 
(operator) 


1 ^ 

product  0  multiplicaud  A 

(remit)  (operand) 


A 

B 

C 

D 

e             ^ 

/ 

§  1.]         DEFINITIONS  AND  GRAPHIC  REPRESENTATION.       253 

So,  even  if  the  operand  be  not  a  vector,  yet  if  it  have  a  re- 
versible quality,  the  vector  oa  may  still  be  taken  as  the  repre- 
sentatice  of  the  operand  ;  and,  since  to  multiply  the  operand  by 
""1  is  simply  to  change  its  quality  into  the  opposite  quality,  this 
multiplication  is  graphically  represented  by  the  reversal  aob, 
while  the  result  is  represented  by  the  vector  ob.  Hence,  in 
what  follows,  the  vectors  used  may  be  either  the  actual  operands 
and  results,  or  merely  their  representatives. 

If  now  there  be  a  continuous  rotary  motion,  as  with  a  spoke 
of  a  wheel,  the  direction  or  quality  of  the  vector  oa  is  alter- 
nately reversed  and  restored : 

E.g.^  a  half  revolution,  one  reversal,  is  multiplication  by  ~1, 

So,  a  whole  revolution,  two  reversals,  is  multiplication  by  ~1 
twice,  z.e.,  multiplication  by  (~l)^  =+1. 

So,  a  revolution  and  a  half,  three  reversals,  is  multiplication 
by  ~1  three  times,  i.e.,  multiplication  by  (~l)^  =~1. 

< — '-4- — t^— >  >  < >  < 


""- — y  no  rev'l.  1  rev'l.  2  rev'l.  3  rev'l. 

So,  multiplying  a  vector  by  ~2  doubles  the  vector  and  reverses 
it ;  multiplying  by  (~2)^  doubles  it  twice  and  reverses  it  twice  ;* 
and  so  on  ;  and  the  like  is  true  whether  the  operand  be  a  vector 
or  not. 
> 


By  such  multiplication  two  distinct  effects  are  produced :  the 
one  quantitative,  the  ordinary  multiplication  of  arithmetic,  which 
consists  in  stretching  the  line  multiplied ;  the  other  qualitative, 
which  consists  in  reversing  the  direction  of  the  line. 

Every  such  multipUer  or  number  may  be  regarded  as  itself 
the  product  of  two  factors  :  its  tensor^  the  quantitative  or  stretch- 
ing factor ;  and  its  versor,  the  qualitative  or  turning  factor. 

If  the  tensor  ^  ^  1 ,  its  effect  is  to  {  g^^f ^en^  the  multiplicand. 

E.g.,  the  number  ~3  is  the  product  of  tensor  3  andversor~l. 
So,      the  number  """f  is  the  product  of  tensor  |  and  versor  '''1. 


254 


OI  AGIN  ARIES. 


[X. 


PARTIAL    REVERSALS.  —  IMAGIXARIES. 

But  during  its  rotation  the  line  has  filled  various  intermediate 
positions  wherein  the  numbers  expressing  its  relation  to  the  unit 
were  neither  purely  positive  nor  purely  negative  numbers : 

E.g.,  in  the  positions 


•^2, 


►'^V> 


<:^::^ 


its  relations  or  ratios  to  a  unit     .  >  are  : 

■•■2,         intermediate,         ~2,  intermediate, 

and  are  represented  thus : 

These  intermediate  numbers  are  imagU 
naries,  or  imaginary  numbers,  and  may  be 
defined  as  numbers,  not  0,  that  are  neither 
purely  positive  nor  purely  negative. 

B}*  way  of  distinction,  positive  and  neg- 
ative numbers,  the  ordinary  numbers  of 
arithmetic  and  algebra,  are  real  numbers. 

It  appears  later  [XIII.]  that  every  imaginary  number  of 
ordinary  algebra  involves  an  even  root  of  a  negative,  and 
arises  from  an  attempt  to  violate  a  condition  of  maximum  or 
minimum :  as  in  seeking  the  base  of  a  right  triangle  whose 
height  shall  exceed  the  hypothenuse. 

The  square  root  of  a  negative  real  number  is  a  pure  imagi- 
nary; all  other  imaginaries  are  complexes. 

E.g.,  the  value  of  -^~4  is  not  +2,  whose  square  is  "^4,  nor  ~2, 
whose  square  is  also  ''■4 ;  it  is  something  different  from  either, 
and  intermediate  between  them  in  character. 

So,  most  roots,  whether  odd  or  even,  and  whether  of  positive 
or  negative  bases,  have  imaginary  values,  as  appears  later. 

THE    SYMBOL   -y/'l. 

The  symbol  -y/~l  denotes  a  number  whose  square  is  ~1 :  i.e.,  it  is 

a  number  such  that  unit  X  -y/'l  X  V~l =u°it  x  ~1.  [I.  §  lOdf .  root 

Hence,  whatever  meaning  is  given  to  multiplication  by  '1,  a 


§  1.]       DEFINITIONS  AND  GKAPHIC  KEPEESENTATION.       255 

consequent  meaning  must  be  given  to  multiplication  by  ^~1 
such  that  two  successive  multiplications  by  -^"1  shall  produce 
the  same  result  as  one  multiplication  by  ~1. 

If  the  unit  be  a  horizontal  line  pointing  to  the  right,  then  the 
product,  unit  X  V"!'  ^^  ^  vertical  line  of  unit  length  pointing 
either  upward  or  downward ;  for  if  the  horizontal  unit-line  be 
first  revolved  to  a  perpendicular  either  way,  then  the  same 
amount  of  further  rotation  will  bring  it  to  the  opposite  hori- 
zontal position.  Here  multiplication  by  -y/^l  consists  in  revolv- 
ing the  multiplicand-line  through  a  right  angle,  either  miti- 
dockwise  or  clockwise. 

So,  when  the  unit  or  operand  is  any  vector  whatever,  ~1  has 
two  distinct  square  roots,  say  i  and  i',  whose  effects  as  multi- 
pliers are  to  revolve  the  line  through  a  right  angle  anti-clock- 
wise and  clockwise  respectively.  Hence  the  effect  of  i'  as  a 
multiplier  is  the  same  as  if  the  multiplicand-line  were  first 
multiplied  by  i  and  then  reversed,  ^.e.,  were  multiplied  by  — i ; 
hence  i'  =  — i,  since  both  numbers  give  the  same  result  when 
multiplying  any  same  unit  [I.  §  1]. 

Since  division  is  the  inverse  of  multiplication,  and  consists  in 
finding  one  factor  when  the  product  and  the  other  factor  are 
given  [I.  §  9],  to  divide  a  given  vector  by  i  is  to  fincj  another 
vector  that,  if  multiplied  by  i,  would  produce  the  given  vector. 
The  quotient  is  the  vector  got  by  revolving  the  dividend-vector 
through  a  right  angle  clockwise ;  for  manifestly,  when  this 
quotient-vector  is  revolved  through  a  right  angle  anti-clockwise, 
i.e.,  is  multiplied  by  *',  the  original  direction  is  restored.  Hence, 
to  divide  any  vector  by  i  is  the  same  thing  as  to  multiply  it  by 
—  i ;  and,  in  like  manner,  to  divide  any  vector  by  —i  is  to  revolve 
it  through  a  right  angle  anti-clockwise,  i.e.,  to  multiply  it  by  i. 

E.g.,  the  unit >  gives  the  products  and  quotients  : 

unit  X  1     unit  X   i     unit  X  "1  unit  X  1     unit  X  —i      unit  X  ~1 

unit  :  1     unit  :—i     unit  :  ~1  unit  :  1     unit  :      i      unit  ;  "1 


Y 


256  niAGIKARIES.  [X. 

So,  the  unit     Nv  gives  the  products  and  quotients : 

unitxl      unitX    i  unit  X  "1  unit  X 1      unitx  — t      unitX'l 

unit  :  1      unit  :-/  unit  :  "1  unit  :  1      unit  :       i      unit  :  "1 


THB   SYMBOLS    -y/    1,    -\/    •^'  ^^^* 

The  operation  of  multiplying  by  ~1  consists  in  reversing  the 
quality  of  the  multiplicand,  and  is  represented  by  one  reversal 
uf  the  line  that  represents  tlie  multiplicand ;  and  the  operation 
of  multiplying  by  -y/'l  is  one  which  if  twice  performed  reverses 
the  quality  of  the  multiplicand,  and  is  represented  by  a  half  re- 
versal of  the  line  that  represents  the  multiplicand. 

So,  multiplying  by  -^~1  is  an  operation  which  three  times  per- 
formed reverses  the  multiplicand,  and  it  is  represented  by  one- 
third  of  one  reversal  of  the  line. 

So,  multiplying  by  ^~1  is  an  operation  which  four  times  per- 
formed reverses  the  multiplicand,  and  it  is  represented  by  one- 
fourth  of  one  reversal  of  the  line  ;    and  so  on. 

The  representatives  of  -^'1,  ^{/-l,  .^"1,  ...  are  the  rotations 
shown  in  the  following  figures,  wherein  lines  of  the  same  length 
as  the  unit  make  with  that  unit  angles  of  Jtt,  ^ir,  ^ir,  •••. 


MULTIPLE    ROOTS. 

But-.-  (-i)^=-i,  (-i)^=-i,  (-ir=-i,  (-iy=-i,  -, 

i.e.,    •.*  1,  3,  5,  7,  •♦.  (any  odd  number)  reversals  has  the  same 
effect  as  one  reversal, 
.-.   ^-1  may  be  represented  by  one-half  of  1,  3,  5,  7,  .•• 
reversals : 


§  1.]        DEFINITIONS  AND  GEAPHIC  REPEESENTATION.      257 

and     -.*  oue-half  of  5,  9,  13,  •••  reversals  are  2^,  4|-,  6 J,  •••  re- 
versals, and  have  the  same  effect  as  a  half  reversal, 
and  one-half  of  7,  11,  15,  •••  reversals  are  3^,  5|^,  7|,  ••• 

reversals,  and  have  the  same  effect  as  IJ  reversals, 
.*.  ~1  has  only  two  distinct  square  roots  in  this  system. 
So,       -^/"l  may  be  represented  by  -J^,  |,  f ,  -I,  •••  reversals  : 
and     • .  •  -J,  ^,  ^,  •  •  •  reversals  =  2 1^,  4|^,  6 J,  •  •  •  rev'ls  =  -J  rev'l, 
and  |,  ^, -^^-,  •••  reversals  =  3,    5,    7,    •••  rev'ls  =  |  rev'l, 

and  -1^,  J^-,      •••  reversals  =  3 J,  5f ,  7|-,  •••  rev'ls  =  -J  revls ; 

.-.  ~1   has  three  cube  roots  represented  b}"  the  curved 
arrow-lines  of  the  figures,  and  but  three. 


iTT 


So,       -^  1  may  be  represented  by  J,  J,  |,  J,  reversals, 
and  ~1  has  four  fourth jroots  represented  by  the  arrow-lines 

of  the  figures,  and  but  four  ;    and  so  on. 


POSITIVE    AND   NEGATIVE    ROTATION. 

Anti-clockwise  rotation  indicated  by  the  figures  is  positive 
rotation,  or  rotation  through  a  positive  angle;  and  clockwise 
rotation  is  negative  rotation,  or  rotation  through  a  negative  angle. 

E.g.,  in  the  third  figure  above  the  two  arrows  indicate  posi- 
tive and  negative  rotation  respectively :  rotation  through  the 
positive  angle  f  tt  and  through  the  negative  angle  —  ^tt. 

The  roots  of  ~1  represented  by  negative  rotation  are  therefore 
identical  with  those  represented  by  positive  rotation  when  taken 
in  reverse  order.  "^ 

£.£/.,   (-i)-i=(-i)l,  (-i)-i  =  (-i)*,  (n)-f  =  (±i)i 

The  reader  may  draw  diagrams  to  illustrate. 


258  IMAGINARiES.  [X. 

MODULUS,   ARGUMENT,   TERSI-TENSOR. 

Every  number  considered  in  algebra,  whether  real  or  imagi- 
nary, ma}'  be  expressed  in  the  form  r*  (~1)**,  wherein  r  is  the 
tensor  or  quantitative  factor  of  the  number,  and  (~1)**  is  the 
versor  or  qualitative  factor.  When  the  number  r(~l)"  operates 
upon  any  vector,  the  result  is  a  vector  of  like  kind,  such  that 
r  is  the  ratio  of  their  leugths  or  magnitudes  and  n  is  tlii>  ratio 
6 :  IT,  which  their  difference  of  direction,  ^,  has  to  two  right  angles. 

If  n  be  an  even  number,  r(~l)"  is  positive  ;  if  odd,  negative  ; 
if  fractional,  some  or  all  of  the  values  are  imaginary. 

The  tensor  r  is  also  called  the  modulus  of  the  number; 
0,=zmr,  is  its  argument  or  versorial  angle;  and  the  number 
r'(  1)"  is  a  versi-tensor. 

Every  abstract  number,  whether  real  or  imaginary,  may  be 
regarded  as  a  versi-ten^or. 

E.g.^  +4,  ~3,  2 1,  —  i  are  versi-tensors  whose  tensors  are  +4, 
+  3,  +2,  +1,  and  versorial  angles  0,  tt,  ^tt,  | tt. 

The  reader  should  clearly  distinguish  between  a  vector  and  a 
tensor  or  versi-tensor.  Vectors  are  lines,  i.e.,  quantities  or 
concrete  numbers,  and  may  represent  any  concrete  numbers, 
operands,  or  results,  that  admit  of  the  same  progressive  change 
of  quality  as  vectors  undergo ;  but  tensors  and  versi-tensors 
are  abstract  numbers,  i.e.,  ratios  or  operators,  and  are  here 
represented  by  the  relations  of  lines  as  to  length  and  direction. 

The  product  of  any  vector  by  a  versi-tensor  is  a  vector  of 
like  kind  ;  tliat  of  two  versi-tensors  is  a  versi-tensor  [§  3]. 

The  properties  of  versi-tensors  are  here  explained  and  de- 
monstrated by  aid  of  the  appropriate  lines  ;  but  they  would  be 
as  true,  though  perhaps  not  as  evident,  if  standing  alone  in 
their  symbolic  form.  It  appears  presenth'  that  versi-tensors 
are  susceptible  of  all  the  ordinary  operations  of  numbers  when 
those  operations  are  properly  defined,  and  that  the  ordinary 
numbers  of  arithmetic  and  algebra  are  but  special  cases  of  these 
more  general  numbers.  The  same  rules  govern  all  sorts  of 
numbers,  and  under  these  rules  all  sorts  of  numbers  may  be 
associated,  and  operated  upon  together  without  confusion  or  error. 


§2.]  ADDITION   AND   SUBTHACTION.  259 

§  2.     ADDITION    AND    SUBTRACTION. 

In  adding  two  or  more  numbers,  two  different  results  ma}-  be 
sought:  (1)  the  arithmetic  sum,  or  sum  total,  wherein  no  re- 
gard is  paid  to  signs  of  quality  ;  (2)  the  algebraic  or  net  sum, 
wherein  the  quality  and  relations  of  the  numbers  are  considered. 

E.g.^  if  a  railway -train  has  run  sixty  miles  east,  and  then 
forty  miles  west  over  the  same  track,  the  total  mileage  is  one 
hundred  miles ;  but  the  distance  it  now  stands  east  of  the 
starting  point  is  but  twenty  miles. 

So,  if  a  sportsman  walk  ten  miles  east,  then  ten  north  and 
ten  west,  he  walks  thirty  miles,  but  is  only  ten  miles  distant, 
and  due  north,  from  camp. 

So,  if  several  forces  not  all  parallel  to  each  other  be  applied 
to  a  body  at  the  same  point,  the  effective  thrust,  their  resultant, 
is  a  single  force  acting  along  a  line  that  may  be  parallel  to 
none  of  them  and  is  less  than  their  arithmetic  sum. 

Two  or  more  vectors  are  added  by  placing  the  initial  point  of 
the  second  upon  the  terminal  point  of  the  first,  the  initial  point 
of  the  third  upon  the  terminal  point  of  the  second,  and  so  on, 
without  changing  their  lengths  or  directions  ;  and  the  vector  sum 
is  that  line  which  joins  the  first  initial  to  the  last  terminal  point. 

E.g.^  of  the  three  lines  ob,  bc,  cd,  below,  the  vector  sum  is 
the  line  od,  whatever  their  length  and  direction  ;  and  this  group 
of  three  lines,  so  far  as  the  effect  is  concerned,  in  carrying  the 
point  from  o  to  d,  maj'  be  replaced  by  the  single  straight  line  od. 


0 

B 

c 

D 

B 

C 

D 

0 

C 

D 

0 

B 

/ 

/' 

/ 

; 

5 

P 

3k 

:// 

3> 

BO  ABO  AGO 


260 


IMAGINARIES. 


[X. 


in  particular,  the  vector  sum  or  difference  of  the  two  perpen- 
diculars of  a  right  triangle  is  the  hypotenuse ;  and  the  vector 
sum  of  two  adjacent  sides  of  a  rectangle  is  a  diagonal. 

E.g.,  in  the  figures  below,    ox  -j-  xp  =  op  and  ox  +  oy  =  op. 


P    P 


O  A      X  X       O  i 

Converselv,  a  line  may  be  replaced  b}'  an}-  group  of  two  or 
more  lines  that  form  a  broken  line  and  have  the  same  initial 
and  terminal  points  as  the  given  line ;  and  the  diagonal  of  a 
parallelogram  ma}'  be  replaced  by  two  adjacent  sides. 

E.g.,  in  the  figures  above  od  may  be  replaced  b}'  oB-f  bc+cd, 
and    op  by  ox  +  xp,   or  ox  -f-  oy. 

The  lines  added  are  vectors  (carriers),  and  their  sum  is  a  vec- 
tor that  reaches  from  the  first  initial  to  the  last  terminal  point. 

So,  when  abstract  numbers,  operators,  are  added  together, 
viz.,  tensors,  versors,  and  versi-tensors,  their  sum  is  a  single 
operator  that,  acting  upon  a  unit  operand,  produces  the  same 
result  as  if  the  several  operators  had  acted  separately  upon  the 
unit,  and  the  results  had  then  been  added  together.  The  sum 
of  the  several  numbers  is  the  same  whatever  vector  be  used  as 
operand :  for  the  vector  sums  got  by  using  different  operands, 
being  obtained  by  like  constructions,  and  so  being  homologous 
lines  of  similar  figures,  as  also  are  the  operands,  bear  like  rela^ 
tions  to  the  respective  operands. 

The  components  of  a  vector  are  any  two  perpendicular  vectors 
of  which  it  is  the  sum.  A  vertical  vector  has  no  horizontal 
component,  and  a  horizontal  vector  has  no  vertical  component. 
An  operator  that  produces  a  vector  perpendicular  to  the  operand, 
or,  more  generally,  that  half  reverses  the  quality  of  anything, 
is  a  pure  imaginary;  and  an  operator  that  produces  an  oblique 
vector  is  a  complex  imaginary. 

E.g. ,  in  the  right  triangle  oxp,  let  oa  be  the  unit  of  length, 
and  let  ox,  xp  be  respectively  parallel  and  perpen- 
dicular to  OA,  and  contain  oa,  in  length,  a,  h  times  ; 


§2.]  ADDITION  AND  SUBTEACTION.  261 

then         the  symbols  a,  6t,  a  +  bl,  stand  for  numbers  that  act- 
ing as  operators  on  the  unit  give  the  lines  ox,  xp,  op. 

If  the  unit  be  horizontal,  if  r  be  the  length  of  any  vector, 
and  0  be  its  inclination  to  the  unit,  then  rcos^  is  the  length  of 
its  horizontal  component,  and  ?*sin^  of  its  vertical  component. 
The  horizontal  component  is  produced  by  an  operator  whose 
tensor  is  rcos^  and  whose  versor  is  1 ;  the  vertical  component 
is  produced  by  an  operator  whose  tensor  is  rsin^  and  whose 
versor  is  i.  Hence  the  oblique  vector  is  produced  by  the  opera- 
tor r  (cos^  +  isin^)  ;  and  the  operator  7*  (  —  1)"  [n  =  ^  :  tt]  is 
equivalent  to  the  operator  r(cos^ +  isin^).  The  first  gives 
the  number  in  its  versi-tensorial  form,  as  the  product  of  a  tensor 
and  a  versor ;  the  second  in  its  complex  form,  as  the  sum  of  its 
two  elements  ;  i.e.,  of  a  real  number  and  a  pure  imaginary. 

If  ic,  yi  be  the  elements  of  any  number,  and  r,  0  the  modulus 

and  argument, 

then         a;=rcos^,    ?/=rsin^,    r=^(a;"+?/-),    ^  =  tan~^(?/ :  a;). 

Anv  number  ricosO  -\-i^m6)  is^    '^^  ^ .1     than  another  num- 
^  '       ^  smaller 

ber,  when  its  modulus  or  tensor,  r,  is  ^  '  "^.^  ,  than  the  modu- 
lus of  the  other :  it  is  ■{  ?^^^  ^^  than  the  other  number  when  its 

real  element^  rcosO,  is-J  ?    '        than  the  real  element  of  the 

other.  The  relations  expressed  by  the  signs  > ,  <  are  inde- 
pendent of  qualit}'  or  direction,  and  they  depend  only  upon  the 
lengths  of  the  vectors  produced,  while  the  relations  expressed 
by  the  signs  >,  <  depend  only  upon  the  horizontal  projections. 

A  number  is  ivJinitesimaU  finite^  or  infinite  when  its  modulus 
is  infinitesimal,  finite,  or  infinite ;  and  the  arguments  of  0  and 
00  are  generally  indeterminate. 

E.g.,  -l±di^l±i,    -l±3i<l±i,    0.i  =  0,    ±00-1  =  00. 

If  the  modulus  r  and  argument  ^  of  a  variable  versi-tensor 
approach  as  limits  the  modulus  ri  and  argument  ^1  of  a  finite 
constant  versi-tensor,  then  cos  (9  =  cos  6)1,  sin  ^  =  sin  6*1,  and  the 
elements  ?'cos(9,  rising  of  the  variable  =  the  elements  riCos,9i, 
rjisin^i  of  the  constant.     Conversely,  if  the  elements  rcos(?, 


262 


IMAGINAEIES. 


[X.  th. 


Y  — 


ri sin d  =  riCOS 01,  riisin^i,  then  the  quotient  r(isia^)  :  rcos^, 
=  /tan^,  =itan^i,  and  6^0i,  and  r  =  ri.  The  constant 
rj (cos  6i + i  sin  6i)  is  then  the  limit  of  the  vanable  r  (cos  ^  + 1  sin  ^) . 

Theor.  1 .   Addition  is  commutative  and  associative. 
(a)    Two  numbers,      •  __-- — ^'^p 

Let  «,  y  be  any  two  numbers  ; 
then  will  x-{-y  =  y -\-x.  vi 

For,  let  OA  be  any  line,  and  let  ox,       /^^^1__— - — jf     x 
OT  be  the  results  of  operating      O  A 

upon  this  line  by  the  numbers  a;,  y ;  complete  the  par- 
allelogram xoY-p ; 
then   •.•  ox  =  YP,  oy  =  xp,  both  in  magnitude  and  in  direction, 

[geom. ,  df.  eq.  lines 
.-.  ox  +  OY  =  ox  +  xp  =  op;  [df.  add.  lines 

and  OY  4-  ox  =  OY  H-  YP  =  op, 

.-.  ox4-OY  =  OY  +  ox,  [Il.ax.l 

I.e.,  a«OA-|-y-OA  =  2/'OAH-a;»OA, 

.-.  x-\-y  =  y  +  x,  Q.E.D.     [df. add. operators 

(6)    Three  numbers. 
Let  a;,  y,  z  be  any  three  numbers  ; 
then  will  x-\-y -{-z  =  x-\- y  +  z  —  'z-\-y  +  x  —  z-\-y  +  x=.'" 
For  let  OA  be  any  line,  and  let 

ox,  OY,  oz  be  the  results 

of  operating  upon  this 

line    by    the    numbers 

a;,  y,  z;    complete   the 

parallelograms    xoy-p 

XOZ-Q,  YOZ-R,  PXQ-S  ; 

then   • .  •  OX  =  YP  =  ZQ  =  KS, 

OY  =  XP  =  ZR  =  QS, 
oz  =  XQ  =  YR  =  PS  ; 


,^-'-p1\ 


and 


I.e.. 


OXH-  OY  +  PS  =  OS,  ox  +  OY+PS 


ox  +  OY  -f-  PS  =  ox  -4-OY  +  PS  =  ••• 


x  +  y'0^-\-z-OK  =  X'OK+y-\-Z'  oa, 
x-\-y-\-z  =  x  +  y-\-z=z  ... 


Q.E.D. 


1,§2.]  ADDITION   AND   SUBTRACTION.  263 

(c)    Four  or  more  numbers. 

1.  The  theorem  is  true  for  two  numbers,  and  for  three,  [(a,  h) 

2.  If  it  be  true  up  to  n  numbers  inclusive,  it  is  true  also  for 

71  +  1  numbers. 
For  let  the  n-\-l  numbers  x^y,  z  -"U^vhe  grouped  and  added 
together  in  any  desired  way,  and  let  the  sum  be  s  ; 
then    •.•  s  is  obtained  by  adding  the  sum,  say  Q,  of  some  of 
these  numbers  to  the  remaining  number,  or  to  the 
sum,  say  k,  of  the  remaining  numbers, 
.-.  s  =  Q-f-R. 
Let  R  be  that  one  of  these  sums  which  contains  the  number  v, 
and  let  t  be  the  sum  of  the  other  numbers  that  make  up  r  ; 

then   •••  neither  Q,R,T  nor  Q-j-T  contains  more  than  71  numbers, 
.'.  in  each  of  them  the  several  numbers  may  take  any 
desired  order  and  grouping  ;  [hyp- 

.-.    S,  =  Q  +  R,  =  Q+T+^ 

=  qj-T+^ [(6) 

=  x-\-y-\-z-\ \-u-{-v 

z=  X -{- y  -}- Z -\ -\-U-\-V.  Q.E.D. 

3.  But  the  theorem  is  true  for  three  numbers  ;  [(5) 
.-.  it  is  true  for  four  numbers,  /  [2 
.*.  for  five  numbers,  for  six  numbers,  and/o  on.     q.e.d. 

Cor.  1.  The  sum  of  two  or  more  numbers  is  the 
sum  of  their  elements.  I 

E.g..)  if  the  first  root  of  each  system  be  taken  ; 
then         ^-1  +  ^-1  +  -^-1 

= ^' + a + i*  V^) + (iV^ + iO 

=  (1+V3)^-1.  ^   '"   ''    A 

CoR.  2.  The  modulus  of  the  sum  of  two  or  more  numbers 
is  not  greater  than  the  sum  of  their  moduli;  and,  if  the  imagina- 
ries  be  unlike,  it  is  less. 

CoR.  3.  If  to  the  minuend  the  opposite  of  the  subtrahend  he 
added,  the  sum  is  the  remainder  sought. 


264  DklAGINARIES.  [X.  tli. 

§  3.      MULTIPLICATION    AND    DIVISION. 

To  multiply  a  concrete  quantity  by  a  versi-tensor  is  to  multi- 
ply the  quantity  by  the  tensor  of  the  multiplier,  and  to  make 
such  reversal  or  partial  reversal  of  the  quality  of  the  result  as 
is  shown  by  the  versor  of  the  multiplier. 

E.g..,  let  ox:OA  be  the  multiplier,  oy  the 
multiplicand,   oz  the  product; 
then         as  to  magnitude,   oz  :  oy  =  ox :  oa, 
and  as  to  quality,  Zyoz  =  Zaox. 

The  product  of  two  or  more  abstract  num- 
bers, versi-tensors,  is  a  single  number  that 
operating  as  multiplier  upon  any  unit  produces 
at  one  operation  the  same  result  as  if  the 
several  versi-tensors  operated  as  multipliers 
in  succession  —  the  first  upon  the  unit,  the 
second  upon  the  first  product,  taken  as  a  new  unit,  and  so  on 
till  all  were  used.  The  product  of  the  several  numbers  is  the 
same  whatever  vector  be  used  as  operand :  for  the  vectors  that 
result  from  using  different  operands  are  obtained  from  these 
operands  by  like  constructions,  and  so  bear  like  relations  to  the 
respective  operands. 

The  quotient  of  one  versi-tensor  by  another  is  that  number 
which  multiplied  by  the  divisor  gives  the  dividend  as  product. 
The  dividend  is  likewise  produced  when  the  divisor  is  multiplied 
by  the  quotient:  for,  as  appears  presently  [th. 3],  the  product 
is  independent  of  the  order  of  the  factors. 

Since,  by  definition  [I.  §8],  the  product  of  two  reciprocals 
is  1,  it  follows  that  the  effect  of  their  successive  operation  as 
multipliers  upon  any  quantity  is  to  leave  the  operand  unchanged  ; 
"ami  that  to  divide  by  a  versi-tensor  is  to  multiply  by  its  recip- 
rocal. 

E.g..,  of  the  two  square  roots  of  ~1,  viz.,  *  and  —  ?,  the 
product  is  unity,  and  they  are  reciprocals :  for  [§  1]  their  suc- 
cessive operation  upon  any  vector  leaves  it  unchanged,  and  to 
multiply  by  either  is  to  divide  by  the  other. 


?,§3.] 


MULTIPLICATION   AND   DIVISION. 


265 


Theor.  2.     If  two  or  more  numbers  he  multiplied  together^ 
the  modulus  of  their  product  is  the  product  of  their  moduli^  and 
the  argument  of  their  product  is  the  sum  of  their  arguments. 
Let  X,  y,  z,  •••  be  any  numbers  severally  equal  to 
r.(-l)^    r'.(-l)"',    r".  (-!)«",    ..., 
whose   moduli   are     r,  r',  r",  •••,    and   whose   arguments   are 

^,  0\  0'\  •••,   equal  to  wtt,  nV,  ti'V,  ••• ; 
then  will  x-y -z-  "•  =  r  -  r' -  7'"  "•  .(-!)»»+«'+«"+•••. 

For  let  OA  be  the  vector  unit,  and  let  a;,  ?/,  2;,  •••,  operating 
on  the  unit,  produce  the  vectors  ox,  oy,  oz,  •••, 
and  let  oa,  ox,  •••  =the  lengths  of  the  vectors  oa,  ox,  ••• ; 
then   *.•  ox       =r(-l)'*.oA,   op  =  r'(-l)"'-6x,  p 

.-.  OP        =r'(-l)**'-r(-l)".0A; 
but     *.*  ox       =z=r'0A,    and    op==r''OX, 
and     •••  Zaox  =  ^,   and    Zxop  =  ^', 

.*.  OP        =r7''-0A,  and  Zaop  =  ^  +  ^', 
.'.OP  may  be  produced  by  acting  on  oa 
with  the  single  operator  whose  ten- 
sor is  rr\  and  whose  versorial  angle 
is  ^  +  6',  or  whose  versor  is  (~1  )'*"•'"'. 
.-.  r'(-iy •  r{-lf ' OA  =  rr'(~l)"+"'  •  ol, 
.-.  r'{-iy' .  r(-l)«        =  rr'{-iy+^'.     q.e.d.  [df.  product 
So,        if  OQ  be  produced  when  the  operator  whose  tensor  is 
r",  =  oz  :  OA,  and  whose  versorial  angle  is  0",  =aoz, 
acts  upon  op  ; 
then    •••  OQ  is  also  got  when  a  single  operator,  with  a  tensor 
rrV",  =  oq:oa,    and  an  angle    ^  +  ^'+^",  =  aoq, 
acts  on  OA, 
.-.  r"{-iy" '  r'(-l)"' .  r{-lY -  oI  =  rrV" (-1)"+"'+""  •  oI, 
.-.  r"(-l)"".r'(-l)'*'.r(-l)^=:r?V(-l)«+"'+"".       q.e.d. 

Cor.  If  one  number  be  divided  by  another^  the  modulus  of 
the  quotient  is  the  quotient  of  the  moduli^  and  the  argument  of  the 
quotient  is  the  argument  of  the  dividend  less  that  of  the  divisor. 

In  particular^  of  a  7iumber  and  its  reciprocal,  the  moduli  are 
reciprocals  and  the  arguments  are  opposites. 


266 


IM  AGIN  ARIES. 


[X.  ths. 


Theor.  3.    Multiplication  is  commutative  and  associative. 
Let  a,  2/,  2  •••  be  any  numbers,  severally  equal  to 


then 

and 

and 

and 

and 


r.(-l)%  r'.  (-!)«',  r".(-l)"'' 
the  product  x-y-z-  '..=zvr''r"- 
the  product  «•  2/*  z ••••=?• 'F^^' 


,n+ni+n"+... 


•(-1)' 


?H=?i'4-n"  +  -..  =  n+n'-{-n"-{-"' ; 

so  for  any  other  order  or  grouping,  [II.  ths.  1,3 

.*.  the  product  a; •  2/ •  ^ —  is  the  same  whatever  the  order 
and  grouping  of  the  factors.  q.e.d. 

Theor.  4.  Multiplication  is  distributive  as  to  addition, 
(a)  The  product  of  the  sum  of  two  numbers  by  a  third: 
Let  a,  y^  z  be  any  three  numbers  ; 

then  will  z •  x  +  y  =  Z'X-\-Z'y.  ""^''7^ 

LetOA  be  any  vector  unit,  and 

let  a;,   y^  z,   operating 

on    the   unit,   produce 

ox,    OY,    6z^    and    let 


>s 


Q-" 


r,  r',  r" ;  6,  e\  0"  be 
the  moduli   and   argu- 
ments of  X,  y,  z. 
Complete  the  parallelogram 
xoY-p ; 
then         OP  =  ox  +  oY. 

Turn  ox,  OY  by  the  angle  0'\  and  stretch  them  in  the  ratio  r", 

making  oq  =  2  •  ox,  or  =  2;  •  oy. 
Complete  the  parallelogram  qor-s  ; 
then    • .  •  OQ  :  ox  =  OR :  OT,   and   Z  xoQ  =  Z  yor,  [constr. 

.-.  OxOYP  is  similar  to  Oqors, 
.-.  Zpos  =  ^"  and   os  =  0-op; 
and     • .  •  OS  =  oQ  +  oF, 

.-.  2;-op  =  2-ox  +  z-oy;   i.e.,  z-ox +  0?=  2;-ox  +  z«oy, 
.'.  z 'X-\-y' OA  =  Z'X-OA-\-Z'y'OA  =  Z'X-^Z'y'OA, 
-.-.  the  product  z-x  +  y  =  the  product  z-x-^z-y.      q.e.d. 


3, 4,  §3.]  MULTIPLICATION   AND  DIVISION.  267 

(6)    The  product  of  the  sum  of  three  or  more  numbers  "by 
another : 

Let  x^y^z^  •••  be  three  or  more  numbers,  and  v  another ; 


then  will  v-a^  +  iZ  +  zH —v-x-^-v-y  ■\-V'Z-\ . 


For      V'X-\-y-\-z-\ —  V'X-\-v.y-\-z-\ [(a) 

•=V'X-\-V'y  -{-V'Z^, 

=  '\)'X-{-V'y-\-V'Z-\''d'~ 

^V'X-^V-y  -\-V'Z-\ .    Q.E.D. 

(c)    Tlie  product  of  two  or  more  polynomials : 

Let  ic  +  y +  «  + ...,   a;'+y'+2!'H ,   be  two  polynomials  ; 

then         x-\-y-{-z-\ •  ic'-f-2/'+2'H 


+  y-a?'  +  2/^  +  g^  +  - 
+  2. a;'4-2/'+z'+ •••  +  ••• 
=  a;-a;'4-a;-2/'4-a;.z'-}-  ••• 

4-2-a;'+0-2/'+2-3'H +  •••.  q.e.d. 

So,       if  this  product  be  multiplied  by  a  third  polynomial 
x"-\-y^'-\-z"-\ ,  a  fourth,  and  so  on. 

Cor.    The  product  of  two  or  more  complexes  is  the  product  of 
the  sums  of  their  elements  used  as  polynomials. 

Let  X,  y  be  any  complexes  such  that  x=p  +  qi,  y=:p'-\-  qH  ; 
then  will  x  •  y  z=p-^qi  'p'-^q'i=pp'—  qq^-\-  i  (pq'-hp'q)  • 
So  for  three  or  more  factors. 
Note.     If  x,  yhe  put  in  the  trigonometric  form 
r-  (cos^  +  isin^),   r'-  (cos^'+isin^'), 
then         X'y  =  rr''  [(cos^cos^'— sin^sin^') 
+  i  (sin  e  cos  0'  +  cos  0  sin  0')'] 
=  rr''  [coa{$-te')  +isin(^  + ^')].  [trig. 

So  for  three  or  more  factors. 


268  IMAGINARIES.  tX.  th. 

§4.      POWERS   AND    ROOTS. 

A  •{  negative  ^'"^^^^^^  power ^  [''(~l)*']'*"*j  of  any  versi-tensor 
r{-l)«,  is  the  continued  \  ^^^^^^^^ 

1  ><:  r(-l)'*  ^  r(-l)*».-.m  times, 

=  1  ><  ?•-  >^  [("l)"]*"  [11.  th.3 

=  1  X  r±"»  X  [(-1)"]^'* ;  [df.  int.  pwr. 

I.e.,  it  is  a  single  versi-tensor  that  multiplying  any  vector 

quantity  would  at  one  operation  stretch  and  turn  it 

in  the  same  way  as  if  r(~l)'*  had  {  ,.  .^}      it  m 

times  in  succession ; 

or,  as  if  the  tensor  r**  had  stretched  it, 

and  as  if  the  versor  [("I)"]'*''*  had  turned  it  m  times  as 

far  as  would  the  versor  ("I)**,  and  in  the  ^  ^^™® 
,.       .  V     /  »  1  opposite 

direction. 

E,g.^    let  OA  be  a  unit,  and  let  the  ratio  ox:oa  be  any 

imaginary  x\ 
make       Z  aox  =  Z  xoy  =  Z  toz  =  •••, 
and  ox :  OA  =  OY :  ox  =  oz  :  OY  =  •  •  • ; 

then         the  ratios  oa:oa,  ox  :  oa,  oy:oa,  •••, 

are  the  numbers  o?.  a^,  a^,  •••, 
and  OA,  ox,  oy,  •••,  the  results  of  using  x  as 

a  multiplier  0, 1, 2,  •••  times  upon  oa. 
So,  the  ratios  op  :  oa,  oq  :  oa,  or  :  oa,  •••, 

are  the  numbers  x~^,  x~^,  x~^,  •••, 
and  op,  oq,  or,  •••,  are  the  results  of  using  cc  as  a  divisor 

1,  2,  3,  •••  times. 

p 
A  fractional poiver,  [r(~l)'*]?,  is  the  pih  power  of  any  versi- 
tensor  whose  gth  power  is  r(~l)'»:  i.e.,  its  effect  when  multi- 
plying any  vector  quantity  is  to  stretch  the  multiplicand  in  the 

p  p       p 

ratio  r?,  and  to  turn  it  as  would  the  versor  [("l)"]*,  or  -  times 

as  far  as  would  the  versor  (~1)". 


5,  §  4.]  POWERS  AND  KOOTS.  269 

An  incommervsurahle  power  [r(~l)'*]"'  [m  incommensurable] , 
is  the  limit  of  [?'("1)*']'"',  wherein  m'  is  a  commensurable  vari- 
able whose  limit  is  m:  i.e.,  [?'(~1)**]"*  denotes  the  versi-tensor 

whose  {  °'<"S"1"%  is  the  limit  of  the  ^  °>o<iulu3        ^^(-i).n„. 

>  argument  '  argument       *-  ^     /J 

as  m'  =  m. 

A  ^ .        .         power  is  any  power  whose  exponent  is  <( . 

•  imaginary^  -^  ^  ^  '  imaginary. 

The  effect  of  an  imaginary  exponent  is  considered  later. 

Theor.  5.    The  modulus  of  any  real  power  is  the  like  power 

of  the  modulus  of  the  base,  and  the  argument  of  the  power  is  the 

product  of  the  argument  of  the  base  by  the  exponent  of  the  power. 

Let  ^(~1)"  be  any  number  whose  modulus  is  r  and  argument 

6,  =  wtt  ;  and  let  m  be  any  real  exponent ; 

then  will  [r(-l)"]"*  =  9''"(-l)'"% 

wherein  r*  is  the  modulus  of  the  product,  and  m6,  =  mn-rr,  is 
its  argument. 

(a)  m  a  positive  integer; 

then         [^("1)'*]'"  =  Ix  r("l)''X  r(-l)«...  m  times 

[df.  pos.  int.  pwr. 
=  r'r'-'m  times •  ("1 ) "+'*"• "" "'"«'  [th.  2 

=  r'"(~l)""*.  Q.E.D. 

(b)  m  a  negative  integer,  say  —  p  ; 

then    •.•   [r(-l)~]-^=l:  [r(-l)"]^  [df . neg. int. pwr. 

=  l:r^(-l)"^  [(a) 

=  r-^(-l)-"^  [§3  th.  2  cr. 

•*•  [.r(-iyy  =?''"(-l)'"\  Q.E.D. 

(c)  m  a  positive  or  negative  fraction  - ;  p,  q  integers; 

then   •.•  [r?(-l)?]*  =r(-l)%  [(a) 

...  r^-(-l)?       =  [r(-l)"]s 

...  rf(-l)f-     =[r(-l)'^]l; 
I.e.,  [r(-l)"]"»  =r'"(-l)'"".  q.e.d. 


270  tMAGINAEIES.  [X.  tlis. 

(d)    m  an  incommensurable; 

Let  m'  be  a  commensurable  exponent  whose  limit  is  m ; 
then   *.•  ?•"'=  lim  tensor  r^' 

and     •••   (~l)'**  =  lim  versor  (~1)"*',  [df .  incom. pwr. 

.-.  r^(-l)«'"  =  lim  versi-tensor  r^X"!)"*'  [§  2 

=  lim  [r(-l)'']"»'  [a,  6 

=  [r(~l)"]'*.  Q.E.D.       [df.  incom.  pwr. 

Theor.  6.  Every  finite  number  has  k  distinct  ktJi  roots,  and  no 
more,  whose  moduli  are  all  equal  and  whose  arguments  are  eqici- 
different. 

Let  r(~l)'*  be  any  finite  number,  a  the  one  real  positive  value 
of  ^r,  m  any  integer;  [VIII.  th.  13 

n+2m 

then  •.•  [a(-l)"*-]*  =  r(-l)"+2«  =  ^(-l)«^  [th.5 

.•.  the  several  roots  sought  are : 

n--4  w-2  n  n+2  w4-4 

...,  a(-l)  *  ,   a(-l)  »  ,   a(-l)S  a(-l)  *  ,   a(-l)  *  , 

n-f2m 

•  ••,  all  of  the  form  a(~l)   *   , 
wherein  a  is  the  modulus  of  all  the  roots  alike, 

,  (?i  — 4)77      (n— 2)7r     TlTT      (n-f-2)7r      (n-f-4)7r 

A/  fC  K  fC  hi 

are  the  arguments ; 
which  differ  from  each  other  by  a  A;th  part  of  2  tt. 
But  only  k  of  these  arguments  have  distinct  effects,  viz. : 
7l7r     (n  +  2)7r      (n+4)7r  ^^^  (n+2  »  fe-l)7r  . 

for      •••  the  other  arguments  differ  from  those  here  named  by 
entire  revolutions, 
.*.  the  corresponding  roots  are  the  products  of  these  roots 
b}'  even  powers  of  ~1,  and  are  identical  with  them. 
And  no  number  with  modulus  not  a,  or  argument  not  embraced 

in  the  list  above,  can  be  a  root. 
For  the  kth  power  of  any  positive  number  not  a  is  not  r, 
and  the  product  of  an}-  other  argument  by  k  is  not  (7i-}-2wi)n-. 


OF 
CAUFOf^ 

6,  6,  §  4.]  POWERS  AND  EOOTS.  271 

Note  1 .  Of  a  positive  real  number  the  kth  root  taxes  the  form 

2rn  l+2m 

a(~l)  *  ;  of  a  negative  real  number,  the  form  a(~l)   *  . 

Note  2.    In  the  trigonometric  form  the  theorem  is  written 

k/  /-i\n        r       (n-\-2m)7r  ,   .   .    (n -{- 2  m)  7r~\ 
-^r{  !)'*=«   cos  i — — i_+^sm-^^ — — ^L 

CoR.  1.    Every  finite  number  has  +k  distinct -th  powers  and 

no  more,  their  moduli  all  equal,  and  their  arguments  equidifferent. 
[/i,  k  any  integers  prime  to  each  other. 

For     • .  •  any  -th  power  is  the  7ith  power  of  a  A:th  root, 

[df.  frac.  pwr. 

and    •••  there  are  k  such  A:th  roots,  whose  common  modulus 

is  a,  [above 

and  whose  arguments  all  differ  by  multiples  of  ~,  less 

than  27r, 

27r 
say  any  two  of  them  by  ^  — -  ;  [^  any  integer  <  k 

k 

.*.  the  corresponding  arguments  of  the  -th  power  differ  by 

h  times  a  •  — ^,  =  ^  •  2  tt, 
-^     k'       k 

and     *.*  k  does  not  measure  gh,  being  >  g  and  prime  to  h, 

.*.  this  difference  of  arguments  is  not  a  multiple  of  27r, 

.*.  all  A;  of  the  -th  powers  are  distinct  in  value  ; 
k 

and    • .  •  their  k  arguments  all  differ  by  multiples  of  —5 

A/ 
.*.  when  taken  in  order, after  rejecting  all  entire  multiples 

of  2  TT,  each  differs  from  the  next  by  —  ; 

A/ 

i.e.,  the  arguments  of  the  powers  are  equidifferent.    q.e.d. 


Cor.  2.  (a)  If  a  commensurable  exponent  m'  approach  some 
limit  m,  whether  commensurable  or  incommensurable,  then  every 
value  of  the  power  [r(~l)°]'^'  approaches  some  value  of  [r(~l)°]™ 
as  a  limit. 


272  IMAGINARIES.  [X.  ths, 

(b)  If  m  be  incommensurable^  the  argument  of  the  power 
[r(~l)°]'°  may  be  indeterminate. 

For     •.*   (a)  the  common  modulus +r^' of  all  values  of  [r(~l )"]**' 
approaches  as  a  limit  the  modulus  ^r*"  of  [r(~l )"]'*, 

and     •.*  the  argument    ??i' (n  + 2A;)7r    of  any  particular  value 
of  [>•("!)"]"*'  approaches  as  a  limit  the  argument 
m(7i  +  2k)  TT  of  that  corresponding  value  of  [r(~l  )'*]'" 
which  is  in  the  same  series, 
.*.  every  value  of  the  power    [?*(~1  )**]'"' 

approaches  some  value  of  [?*("!) "J"*  as  a  limit,  q.e.d. 

And   -.*   (6)  as  the  commensurable  m'  approaches  the  incom- 
mensurable limit  m,  the  successive  convergents  have 
larger  and  larger  denominators,  [continued  fractions 
.'.  the  number  of  distinct  values  of  the  m'th  power  in- 
creases without  limit  as   m^=m\ 

and     •.*  for  any  value  of  m'  these  numerous  values  of  the 
power  have  their  arguments  equidififerent, 

.  .  as  m'=  m     the   arguments  of    consecutive  values  of 
the  power  approach  one   another  more   and   more 
closel}', 
and  in  the  limiting  case,  when  the  exponent  is  the  incom- 

mensurable m,  the  argument  of  the  power  may  be 
regarded  as  quite  indeterminate,  i.e.,  as  continuous. 

Q.E.D. 

Note.  By  convention,  however,  the  values  of  an  incom- 
mensurable real  power  of  a  real  positive  base  are  often  re- 
stricted to  the  single  real  positive  value. 

So,  by  convention,    every   power  of   the  Napierian  base   e 
[XII.  th.  28,  ap.  4,  cr.]  is  restricted  to  its  real  positive  value, 
though  the  powers  of  the  equivalent  number  2.71828  •••  are  not 
so  restricted  ; 
i.e.,        -^e=l-UST2"'  only;  but  V2-71828..-=  ±1.64872.... 


6,  7,  §4.]  POWERS  AND   ROOTS.  273 

Theor.  7.    The  product  of  like  powers  of  tivo  or  more  bases 
is  the  same  power  of  the  product  of  the  bases. 

0  9' 

Let  the  bases  be  r{^l)^^  r'(~l)^  •••,  whose  moduli  are  r,rV*-? 
and  whose  arguments  are  ^,  ^',  •••  ;  and  let  m  be 
any  real  exponent : 

then   •.•  of  the  powers   [r(-l)if]"',    [r'(-l)F]"',  ..., 

the  moduli  are  r",  r'"*,  •••, 

and  the  arguments  are    m9,  m$\  ••• ; 

.'.  of  the  product  of  powers    [r(-l)'r]"'.  [r'(-l)i"]"»... 
the  modulus  is     7'™ •  r ''"••• ,  =  (rr'- ••)"', 
and  the  argument  is   m^4- w^'-f- •••,  =  m  (^ +  ^'H- •••)  ; 

and     •.*  the  product  of  the  given  bases  has  modulus  rr'*-* 

and  argument  6.-\-0^-\ , 

.'.  its  mth  power  has  modulus    (rr'"-)'" 

and  argument  m(^  +  6'+  •••)  ; 

t.e.,  the  product  of  mth  powers  of  the  bases,  and  the  mth 

power  of  the  product  of  the  bases,  have  the  same 
modulus  and  argument,  and  are  equal,      q.e.d. 
Cor.    Tlie  quotient  of  like  powers  of  two  bases  is  the  same 
power  of  the  quotient  of  the  bases. 

Note.     When  the  exponent  ??i  is  commensurable,  and  the 
arguments  ^,  ^',  •••  of  the  given  bases  are  so  related  that  the 
values  of  their  sum  0-\-0'  -\-  -"  cannot  differ  from  one  another 
except  by  certain  of  the  multiples  of  2  tt,  it  may  happen  that 
the  power  of  the  product  or  quotient  has  more  distinct  values 
than  the  product  or  quotient  of  the  powers.  [comp.VIII.  th.  2  nt.  2 
E.g. J    let  two  given  bases,  and  their  products,  be  1  + 1,  2i, 
—  2  +  2  /,  whose  moduli  and  arguments  are  : 
V2,  i7r  +  2^7r;   2,  ^7r'+27c7r;    y8,^7r-{-2l7r, 
wherein   /i,  Zj,  1  =  any  integers,  positive,  negative,  or  zero ; 
then,        in  general,   (1  +  i)"*  •  (2i)'"  =  ("2  +  2i)'»  ; 
I.e.,         every  value  of  either  member  is  a  value  of  the  other 

member, 
for  the  modulus  of  either  member  is  +82, 

and  the  argument  of  either  member  is  m  •  (Itt  +  Z«27r), 

wherein   I,  =h-}-k,  is  any  integer  whatev.er. 


274  ESI  AGIN  AEIES.  [X.  ths.  7-9, 

But      if  m  =  1^,  and  if   it  happen  that  in  the  investigation 
from  which  the  bases  1  -f-z,  2i,  ~2  +  2i  arise,  the 
2 1  is  got  as  the  square  of  the  1  +  z,  while  the  ~2+2t 
presents  itself  independently, 
then   •.'  k=2h^  while  I  remains  unrestricted, 
.*.  the  argument  of  (1  +*)'"•  (2  4')"*  is 

iU7r4-(/i  +  2/i)27r],    =i7r  +  7i.27r, 

while        the  argument  of  (~2  -\-2i)"'  is 

i(f7r-fZ.2,r),  =}7r+?.|7r; 
i.e.,  the  product  of  the  powers  has  only  one  value, 

while        the  power  of  the  product  has  three  distinct  values. 

Theor.  8.  TJie  product  of  two  powers  of  any  same  base,  in 
any  same  series^  is  that  power  of  the  base  whose  exponent  is  the 
su7n  of  their  exponents,  and  is  in  the  same  series. 

Note.     Different  powers  of  a  base  are  in  the  same  series, 

when  they  arise  from  attributing  to  the  base  the  same  argument 

and  not  arguments  differing  by  one  or  more  entire  revolutions  ; 

i.e.,  when  their  bases  are  identical  and  not  merely  equivalent. 

[comp.VIII.§l,  VIII.  th.  10 
e 
Let  the  base  be  a,  =r(~l)'!-,  whose  modulus  and  argument 

are  r  and  6  ;  and  letp,  q,  •••  be  an\'  real  exponents  ; 

then   '.-  A^,  A',  •••    A^+«+-   have  the  moduli   ?-^,  r',  •••  r^+2+" 

and  the  arguments  p6,  qO,  •••  {p  +  q-\ )6,  [th.  5 

and     •••  r^.r'.--  =  r^+«+-,  [Vlll.th.  10 

and  p^-f-g^+...=  (p-t-g-f....)^, 

.*.  the  product  of  the  moduli  of  aP,  a',  •••  is  the  modulus 
of  A^+«+-, 
and  the  sum  of  the  arguments  of  a^,  a'',  •••  is  the  argument 

of  A^+'+-; 
i.e. ,  the  product  a^  •  a'  •  •  •  =  a^«+-  ;  q.e.d.     [th.  2 

and     •••  the  argument  of  this  product  is  {p-\-q-\ )6, 

and  not  (p  +  g  H h  2k7r)0, 

,'.  the  product  is  in  the  same  series  as  the  factors,  q.e.d. 
Cor.     The  quotient  of  two  powers  of  any  same  base,  in  any 
same  series,  is  a  power  of  the  base  whose  exponent  is  the  differ- 
ence  of  the  given  exponents;  and  it  is  in  the  same  series. 


pr.l,  §4.]  POWERS  AND  HOOTS.  275 

Theor.  9.    A  power  of  a  power  of  any  base  is  that  power  of 

the  base  whose  exponent  is  the  product  of  the  given  exponents. 

e 
Let  the  base  be  a,  =  r(~l)'r ;  and  let  m,  n  be  any  exponents  ; 

then    •.•a'"  has  the  modulus  r"*  and  the  argument  mO, 

.-.   (a"*)**  has  the  modulus  (r"*)**  and  the  argument  n{mB)  ; 

i.e.,  (a"*)"  has  the  modulus  ?-'""  and  the  argument  mnO ; 

but  A"*"  has  the  same  modulus  r™"  and  argument  mnO ; 

.-.   (a"*)"  =  a""*.  q.e.d. 

Note.  If  a  base  b  be  not  identical  with  a*"  but  only  equivalent 
(th.  8  nt.),  and  if  n  have  a  denominator  q\  then  b"*  may  have 
values  not  included  among  those  of  a"*"  ;  and  Theor.  9  may  be 
stated  as  follows : 

Of  any  number  known  merely  to  be  equivalent  to  a  given 
power  of  a  given  base,  any  given  power  includes  among  its 
values  all  values  of  that  power  of  the  given  base  whose  exponent 
is  the  product  of  the  given  exponents.         [comp.  VIII.  th.  4  nt. 

E.g.^    if  ^,  the  argument  of  a,  be  a  +  27i7r, 
and  if  B,  =  A"*  but  ^  a*",  have  argument  mO  +  2A;7r, 

wherein  h^  k  may  take  in  succession  all  integral  values, 

then    •.•  A*"l  has  the  argument  !!?^  •  a  +  ^  •  2 tt, 

q  q 

and     •.•  B?  has  the  argument  !!^  .  a  +  ^^^^^  "^  ^^ •  27r, 

p  p  p 

.'.  B?  takes  every  value  of  a"*?,  but  it  may  be  that  b«  takes 

other  values  besides. 

PrOB.   1.     To   FIND   THE  Wth  ROOT  OF  ANY  REAL  NUMBER,    itt": 

Put  X  for  the  roots  sought;  then: 

0  2 

To  find  the  nth  root  of  a°,  write  x  =a  ("1)^,    a(~l)s, 

a(-l)n,  ...  a(-l/"^;  [th.  5 

i.e.,         write  x  =  B,(cosO -\-\ sin 0),  alcos his*^— -j'  **•• 

1  8 

To  find  the  nth  root  of  —  a",  write  x  =  a  (~1)'»,  a(~l)°,  . 

5  2n-l  . 

a(~l)n,    ...  a(-l)   n    5 

/       TT   .    .    .    7r\         /       Stt    ,   .    .    37r\ 

I.e.,         write  ■KzzzQ.l  cos- -\-\si7i-].    afcos \-\sin — N  •••• 

V      n  nj'      \       n  nj 


276  IMAGIN ARIES.  [X.  pr. 

1.  To  find  the  square  root  of  a? : 

then    •.•  «=  a(cosO  +  isinO),    a(cos|7r  +  isin|7r), 
and     '.*  cosO  =  l,    siuO  =  0;   cos7r  =  --l,    sm7r=0,        [trig. 
.*.  x  =  a,    —a. 

2.  To  find  the  square  root  of  —a^: 

then   •••  a;  =  a(cosj7r-|-isiniJ^7r),  a(cos|7r  +  ismf  tt), 
and     •••  003^77  =  0,  sin |7r=  1 ;  cosf 7r=  0,  sinj7r=  — 1,  [trig. 
.'.  x=  ai,    —ai. 

3.  To  find  the  cube  root  of  a^ : 

then   •••  a:  =  a(cosO  +  isinO),   a(cosf 7r  +  esinj7r), 

a(cosj7r  +  tsin|7r), 
and     •••  cosO  =  l,  sinO  =  0;  cos|7r  =  — |^,  sin  j tt  =  ^-y^S  ; 

cos|7r=-i,  sin|7r  =  -|V3,  [trig. 

.-.  x  =  a,  ja(-H-iV3),  ^a{- 1  -  i-y/3). 

4.  To  find  the  cube  root  of  —  a^ : 

then  ,.*.  a;  =  a(cos^7rH-isin^7r),  a(co8f  7r  +  tsinf  tt), 

a  (cos  ^  TT  +  *  sin  ^tt)  , 
and     *.•  cosj7r=J,    8ini7r=^V3;    cos7r  =  — 1,    sin7r  =  0; 

cos|7r  =  i,    sin|7r  =  --J-V3,  [trig. 

.',  x=ia{l  +  i^B),  —a,    ia(l-iV3). 

5.  To  find  the  fourth  root  of  a* : 

then   *.•  a;  =  a(cos0  4-*sin0),   a(cos}7r  +  tsin|7r), 
-  a(cos^Tr-f- isin|7r),   a(cosf  7r-}-isin|7r), 
and     •.•  cosO  =  l,    sinO=0;  cosj7r=0,    sin^7r=l; 

cos7r=—  1,  siuTT  =  0  ;  cos|7r=0,  sinf  TT  =  —1  ;[trig. 
.*.  x  =  a,   ai,    — a,    — ai. 

6.  To  find  the  fourth  root  of  —  a* : 

then   •••  a;=a(cos:|^7r  +  isinj7r),   a(cosf 7r  +  isinf tt), 
a(cosj7r-f-isinj7r),    a  (cosJ-;r  +  isin  Jtt), 

and     •.*  cosj7r  =  y'i-5    sinj7r  =  ^|-;  cosf 7r=  —  ^|-, 

sinf7r  =  Vi;  cosj7r  =  — Vii    sinJ7r  =  — Vi; 
cosj7r  =  V^)   sin|-7r  =  — Vi;  [trig. 

.-.  a;=ia(V2  +  *V2),    ha(-^2+i^2), 


1,  2,  §  4.]  POWERS   AND  liOOTS.  277 

7.  To  find  the  fifth  root  of  a^ : 

then    •••  a;=a(cosO  +  ismO),    a(cosf 7r  +  ism|7r),  ••., 
and     •••  cosO=l,     sinO  =  0;  cos|7r  =  ^  (V^  ~  l)? 

sin|,r=  iV(10  +  2V5)  ;  •••»  [trig. 

i^[-(V5  +  i)+*V(io-2V5)], 

ia[-(V5  +  l)-iV(10-V5)], 
ia[(V5-l)-iV(10  +  2V5)]. 

8.  To  find  the  fifth  root  of  -  a^ : 

then   •.•  x=a(cos^Tr-\-i8m\Tr),   a(cosf7r  + isinfTr),  •••, 
and     .-.  cos^7r  =  ^(-^5  + 1),  sin|7r=JV(10  — 2^5); ...  [trig. 
...  £c  =  ia[(V5  +  l)+^V(10-2V5)], 

ia[-(V5-l)+iV(10  +  2V5)],  -a, 
ia[-(V5-l-iV10  +  2V5)], 
ia  [( V5  +  1)  -  t  V(10  -  2V5)]. 
And  so  for  other  roots. 

PkOB.  2.       To    FIND   THE    Wth   ROOT   OF   AN    IMAGINARY    a-^M: 

write        r  =  V  (a-  +  b^) ,    0  =  tan'^  (b  :  a)  ; 
then         a  +  bi  =  r  (~1)'^,  =r{cos9 -\-\sin6)\ 

and  (a  +  bi)n  =  rn(~l)n'r,   rii(~l)  htt  ,   ri(~l)  n/r  ,    ..., 

=  Tn{  COS  -  +  1  sin-  L    rs  cos h  i  sm ,  ••• 

V       n  ny'       \  n  n     J 

E.g.,        to  find  the  fourth  root  of  1—^~B: 
then   •.•  a=l,    b  =  -^S,   r=2,   ^  =  |7r,  |7r,  -I^tt,  ^O-tt; 
...  l-V-3  =  2(-l)f,  2(-l)l,  2(-l)¥,  2(-l)¥, 
=  2(cos-|7r  4-i  sin -|7r),       2  (cos  |7r  +i  sin|7r) , 
2(cos^+  isin-y-Tr),  2(cos-2/7r+isiny7r), 
.-.   (l-V-3)i=2^(-l)*,      2i(-l)^  2i(-l)i  2i(-l)^ 
=  2^(cos|-7r  +  isin|-7r),     2^cosf7r+isin|-7r), 

2^  (cos  Jtt  +  *■  sin  Itt)  ,      2 1  (  cos  J/^r  +  i  sin  ^^-tt)  , 
=  1.0299  +  .59451,      -  .5945  +  1.0299i, 
-  1.0299  -  .59451,      .5945  -  1.0299i. 


278  rSIAGINARIES.  [X. 

§  5.    ABRIDGED    REPRESENTATION. 

In  man}'  important  applications  of  the  theory  of  imaginaries 
their  representation  is  abridged  as  follows  : 

A  fixed  point  or  origin  o  is  chosen,  and  a  unit-line  oa  point- 
ing to  the  right  is  taken  as  the  common  operand  of  all  the 
imaginaries  to  be  represented  ;  then,  of  any  imaginar}-  op  :  oa,  op 
is  the  representative  vector^  and  p  is  the  representative  point;  for, 
since  the  operand  ol  is  the  same  for  all  the  imaginaries,  the 
resulting  vectors  or  even  their  terminal  points  are  sufficient  to 
distinguish  one  operator  from  another. 

In  this  abridged  representation,  the  thing  chiefly  present  to 
the  mind  is  the  point  p ;  and  every  number,  real  or  imaginary, 
is  conceived  to  be  written  at  its  representative  point,  in  the  plane 

GAP. 

E.g.^  if  p,  Q  be  the  representative  points  of  any  numbers  p,  g, 
then  p  is  further  than  Q^  f^o^^'ull^^'i^i^  when  p  ^  >5;  the 

middle  point  of  pq  is  the  representative  point  of  h  {p  +  q)  ;  and 
PQ  is  the  representative  vector  of  {q  — i?). 

So,  if  A,  B,  c,  D  be  the  representative  points  of  a,  6,  c,  d 
respectively,  and  if  a -\- c  —  h -\- d^  then  abcd  is  a  parallelogram 
whose  equal  sides  ab,  dc  are  representative  vectors  of  the  equal 
numbers  (6  —  a) ,  (c  —  cZ) ,  and  whose  centre  is  the  representa- 
tive point  of  J  (a  -H  6  H-  c  -f-  cZ) . 

If  a  variable  pass  from  one  value  to  another  by  continuous 
change,  then  its  representative  point  moves  along  some  locus, 
the  path  of  the  variable. 

E.g.^  the  path  of  a  real  variable  lies  in  the  line  oa  ;  the  path 
of  a  variable  pure  imaginar}-  lies  in  the  line  through  o  perpen- 
dicular to  OA ;  the  path  of  a  variable  whose  versorial  angle  is 
constant  is  a  straight  line  through  o ;  the  path  of  a  variable 
whose  tensor  is  constant  is  the  circumference  of  a  circle  whose 
centre  is  o. 


§  6.]  EXAMPLES.  279 

§  6.     EXAMPLES. 

§  1- 

1 .  Assume  any  convenient  linear  unit,  and  plat  the  numbers  : 

0(-l)«;    1(-1)S  -l(-l)-';    2(-l)^  -2(-l)-^ 
3(-l)',-3(-l)-^  4{-iy,-i{'l)-*;  o(-l)»,-5(-l)-». 

2.  So,  the  two  values  of  -^4  and  of  ^^"4  ;  the  three  values  of 

^27,  and  of  -^-27 ;  the  four  values  of  ^256,  and  of 
^-256  ;  the  five  values  of  -^3125,' and  of  -^-3125  ;  the 
six  values  of  -J/216,  and  of  ^"216  ;  the  eight  values  of 
^a«  and  of  -^'a^ ;  the  ten  values  of  ^a^  and  of  ^^'a}'' ; 
the  twelve  values  of  ^a^  and  of  ^~a}^,  wherein  a  is  any 
positive  real  number. 

§2. 

3.  Find  the  moduli  and  arguments  of  the  following  numbers, 

given  by  their  elements  : 

i+iV"3,  i-iV"3  ;  1+*,  1-M    2V3+2Z,  2V3-2i; 

Vo  +  1  ±  ^V(10  -  2  V5) ,   -  V^  -1  ±  V(10  -  2  V5) , 
wherein  i(V5  +  l)  =  ^os36°,  ^■^(10'-2^5)  =  sm3G°  ; 

V5-1±H/(10  +  2V5),  -V5+1±^V(10  +  2V5), 
wherein  i(V5-l)=cos 72°,  iVC^^+^V^)  =  sin72°. 

4.  If  a,  h  be  any  two  real  numbers,  show  that  the  modulus  of 

the  complex  imaginary  a±hi  is  V(<^^+^0>  ^^^  ^^^  argu- 
ment, tan  ~\  ±  6  :  a) . 

5.  Assume  any  convenient  linear  unit,  and  plat  the  following 

numbers  ;  add  them,  compute  the  moduli  and  arguments 

of  the  several  sums,  and  plat  those  sums. 

2i^,2i^;  2i2,2i^2r^;  3^^3^^3^2;  3i-2,3r^3r'^\ 

6.  Draw  an  equilateral  triangle  abc,  assume  the  base  ab  as  the 

linear  unit,  then  show  what  numbers  will  produce  the  sides 
AB,  BC,  CA,  and  find  their  elements. 
Add  these  numbers  and  show  that  their  sum  is  0. 

7.  So,  for  the  square,  the  regular  pentagon,  the  regular  hexa- 

gon, the  regular  octagon,  the  regular  decagon,  the  reg- 
ular dodecagon,  assuming  an}^  side  as  unit. 


280  IMAGINARIES.  [X.  §  6. 

§3. 

8.  Find  the  products,  and  show  graphically  that  they  are  real,  of 

2  +  3i,  2-3i;  a-\-bi,  a  —  bi;  ^3-h^■^y6,  -^S  -i^5; 

9.  Find  the  products  of  the  sets  of  numbers  in  Exs.  5,  6. 

10.  Multiply  a-\-bi  by  c-j-di,  and  show  that  the  product  takes 

the  form  p  +  gi,  wherein  p  =  ac  —  6d,  q  =  6c  +  ac? ;  and 

that  r  =  V(«'+  b')  (0^+  (f-),  and  6  =  tan"^  bc±ad^ 

ac  —  bd 

11.  Divide  a.-\-bi  by  c  +  c^i,  and  show  that  the  quotient  takes 

the  form  p  +  Qh  wherein  p  =  ^^P^j,  q  =  ^^=^f  ;    and 

that  r  =  ^1^,  and  ^  =  tan"^  ?^^^1^. 
\c2  +  cP  ac-f6c2 

12     Divide  ^  "^^^'  •  9lIiM-    ^  +  6i  .  a  —  bi^    a  —  bl  ,  CT-f-6i 
a  —  bi    a  +  6i '   c  -f  di     c  —  di '    c  4-  cii  *  c  —  d^ 

13.  Multiply 

2l*+2^'by2^*4-2^^;3^^+3^^+3i«by3^-2+3^~^^-3^•~''A 
Plat  the  products  ;  find  their  elements,  moduli,  and  arguments. 

14.  Express  two  or  more  versi-tensors  as  complexes,  and  using 

them  in  that  form,  show  that  the  multiplication  of  such 
complexes  is  both  commutative  and  associative. 
§4. 

15.  Resolve  the  numbers  given  in  Ex.  2  into  their  elements. 

16.  Find  the  product  of  the  two  values  of  ^4,  and  of  -^^"4 ; 

of  the  three  values  of  -^27,  and  of  .^"27  ; 
of  the  four  values  of  -^256,  and  of  ^"256  ; 
of  the  five  values  of  ^3125,  and  of  ^/-3125. 
Write  the  several  factors  in  the  three  forms  r(~l)'*,  r-i^*, 
and  a  -f-  bi ;  and  show  that  the  products  so  found  are  the 
same  whichever  form  be  used. 

17.  Find  the  powers  and  roots  as  indicated : 

[2  +  i(V^  +  l)  +  iV(10-2V5)]^   (3i-f3i2)i 


XI.  §  1.]  STATEMENTS.  281 


XI.     EQUATIONS. 

^P*  For  definition  of  the  words  equation^  identity^  inequality^ 
statement^  memhei'^  and  copula^  see  I.  §  5. 

§  1.     STATEMENTS. 

One  statement  is  a^      ^.     ^  condition  ot  another  when,  if 

the  first  be  -{  . '      '  the  other  is  also  ^  . 

'  true,  '  true. 

Two  statements  are  equivalent  if  one  be  both  a  necessary  and 

a  sufficient  condition  of  the  other,  i.e.,  if  they  be  false  together 

and  trne  together. 

Two  or  more  statements  are  {  .'  ..,,   when,  if  some  of 

'  incompatible  ' 

them  be  true,  the  others  must  be  ^  ^  ,    \  independent  when, 

whichever  of  them  be  true  or  false,  the  rest  may,  just  as  well. 

be  true,  or  be  false. 

rr,,  I  necessary  conditions  ,  associated 

There  are  n  <        .      .y..  amonff  m  <  .  ..-,■, 

'  contradictions  ^        '  incompatible 

statements  when  some  m  — n  of  these  statements  are  independent, 
and  if  these  be  true,  the  remaining  statements  are  -{  ^ ,,  .  ,   * 

E.g.^  the  equation  x  —  S  is  equivalent  to  the  equation  2x  —  6, 
and  it  is  a  sufficient,  but  not  a  necessary  condition,  of  the  in- 
equality  aj<4. 

The  three  statements  are  associated  and  have  two  necessary 
conditions  among  them,  since,  if  the  first  be  true,  so  are  both 
the  others. 

The  last  two  are  necessar}'  conditions  of  the  first ;  the  third 
is  not,  but  the  second  is,  a  sufficient  condition  of  the  first. 

So,  the  inequalities  a;<?/,  y<z,  z<,x  are  incompatible,  in- 
volving one  contradiction ;  for,  though  any  two  of  the  state- 
ments may  be  true  or  false,  or  one  be  true  and  the  other  false, 
yet  if  two  of  them  be  true,  the  third  must  be  false. 

So,  the  equations    x-\-y  =  2,  x-^2y  —  S.  x  +  oy—^^   are 
.  associates,         v,     *  o-        J  "^cessary  condition  oj^^j.^.  them 
'  incompatibles,    ^^^^°  ^^^ '^  contradiction  ^ 


282  EQUATIONS.  [XI. 

§2.     SOLUTION   OF   EQUATIONS.— UNKNOWNS. 

The  letter  or  letters  for  which  particular  values  are  sought 
that  shall  make  true  the  statements  contained  in  the  equations 
are  the  mikfwicn.,  as  distinguished  from  the  other  elements, 
that  are  given  and  now  called  known  elements ;  and  the  solu- 
tion of  an  equation,  or  system  of  equations,  consists  in  making 
such  transformations  therein,  as,  while  the  equality  of  the  mem- 
bers is  preserved,  and  the  relations  between  the  elements  are 
unchanged,  shall  result  in  giving  the  values  of  the  unknown 
elements  in  terms  of  the  known  elements.  The  values  so  found 
are  the  roots  of  the  equation  or  system  of  equations  ;  and  the 
test  to  be  applied  to  them  is  to  replace  the  unknown  elements 
by  these  values,  and  see  if  they  make  the  equations  identities. 

E.g.^  of  the  equation   2a;=  4    [a;  unknown]    2  is  a  root, 
•••  2-2  =  4,    a  numerical  identity.  [df .  root 

So,  of  the  equation  ar—bx-{-G  =  0  [a; unknown]  2,  3  are  roots, 
•.•  22-5.2  +  6  =  0,    32-5.3  +  0  =  0. 

So,  of  the  equation   a^  =  a^  [_x  unknown] , 

a,    ia(  — 1+ i-y^3),    ia(  — 1-1^3)    are  roots, 
•.•  a^=a\  [ia(-^l+iv3)]'  =  a3,  [ha(-l-i^3)Y=a\ 

Equations  that  involve  the  same  unknown  elements,  and  are 
satisfied  by  the  same  values  of  them,  are  simultaneous  equations  ; 
and  those  values  are  simultaneous  values. 

E.g.,  it  the  equations  2a; +  5?/ =19,   6x  —  Sy  =  S   [a;,  ?/ un- 
known] be  simultaneous,  2,  3  is  a  pair  of  roots, 
•.•  2.2  +  5.3  =  19,     and     6.2-3.3  =  3. 
So,  of  the  simultaneous  equations    x  —  y  =  5,    x^-\-y^=lS 
2,-3  ;   3,-2  are  pairs  of  roots  ;  but  not  2,-2  ;   3,-3. 

So,  if  two  plane  curves  be  expressed  b}*  two  equations  involv- 
ing two  variables,  for  the  points  of  meeting  both  curves  have  the 
same  coordinates,  and  for  these  points,  but  for  no  others,  the 
two  equations  are  simultaneous. 

The  roots  of  an  equation  are  sometimes  called  its  solution. 


§§2,3.]  DEGEEE   OF   EQUATION.  283 

§3.  DEGEEE  OF  EQUATION. 
If,  without  extracting  roots,  an  equation  involving  one  un- 
known element  be  so  transformed  that  both  members  are  entire 
as  to  that  element,  the  degree  of  the  equation  is  the  degree  of 
that  term  wherein  the  degree  of  the  unknown  element  is  highest. 
If  the  equation  contain  QYQvy  power  of  the  unknown  element, 
from  the  highest  to  the  zero  power  inclusive,  it  is  a  complete 
equation  ;  if  not,  it  is  incomplete. 

E.g. ,    the  equation    dbx  =  cd-^ef   \_x  unknown]  is  of  the  first 

degree,  a  simple  equation. 
So,       the   equation    y^+3y  =  4d      [2/ unknown]   is  of   the 
second  degree  ;  it  is  a  complete  quadratic  equation; 
but  the  equation     y^  =  49  is  an  incomplete  quadratic  ; 

and  the  equation     if  -\-Oy  —  Ad   is  a  complete  quadratic. 

So,       the  equation     ?-^  +  5r^4-5r  =  426    [?-  unknown]  is  of 

the  third  degree  ;  it  is  a  complete  cubic  equation. 
So,       the  equation    k*-{-12  Jc^  +  60 k-  +  95  k  =1230     [k  un- 
known] is  of  the  fourth  degree ;   it  is  a  complete 
biquadratic  equation. 
So,       the  equation   s  =  a(r'*—  1)  :  (r  —  1)  is  of  the  71th  de- 
gree if  r  be  the  unknown  element ; 
of  the  first  degree,  if  s  or  a  be  the  unknown  element ; 
an  exponential  equation  if  n  be  the  unknown  element. 
An  equation  may  contain  one  unknown  element  or  more. 
E.g.^   the  equations  above  have  each  one  unknown  element; 
but  the  equation    ax--\-'2 lixy  -^by""  -\-2gx-\-  2fy  -f  cZ  =  0    is 

a  complete  quadratic  with  two  unknown  elements ; 
and  aa?  +  by^  -f-  cz^  -\-  2fyz  +  2 gzx  +  2 hxy  +  2lx-i-2my 

-\-2nz-{-d=:0    [_x,  y,  z  unknown]  is  a  complete  quad- 
ratic involving  three  unknown  elements. 

If  an  entire  equation  involve  two  or  more  unknown  elements, 
the  degree  of  the  equation  is  the  sum  of  the  exponents  of  the 
elements  in  that  term  in  which  their  sum  is  greatest. 

E.g.^  the  equation  Sxy^-^2x^-\-y^-i-x-{-y-^27  =  0  is  a  cubic, 
but  not  complete. 


284  EQUATIONS.  [XI.  ths. 

§4.     GENERAL  PROPERTIES. 

Theor.  1 .  If  to  both  members  of  an  equation  the  same  num- 
ber be  added,  the  roots  of  the  equation  are  not  changed  thereby. 

Let  p  =  Q  be  any  equation,  and  n  any  number  ; 
then  are  the  roots  of  equations  p  =  q,  p4-n  =  q+n,  identical. 
For     •.*  p  +  N  =  Q-hN   when   p  =  Q,    and  then  only, 

.-.  every  root  or  set  of  roots  of  the  equation  p  =  Q,  sat- 
isfies the  equation  p  -f-  n=  q  +  n,  and  conversely  ; 
I.e.,  every  root  of  either  equation  is  a  root  of  the  other. 

Q.E.D. 

CoK.  1.  If  any  term  be  transposed  from  one  side  of  an  equa- 
tion to  the  other  and  its  sign  reversed  the  roots  of  the  equation 
are  not  changed  thereby. 

E.g..,    the  roots  of  the  equations 

aar' +  6a;  +  c  =  0,     ax^  +  6aj  =  — c,   are  identical. 

Cor.  2.  If  the  signs  of  all  the  terms  of  an  equation  be  changed 
the  roots  of  the  equation  are  not  changed  thereby. 

Theor.  2.    If  both  members  of  an  equation  be  multiplied  by 

any  same  number,  not  a  function  of  the  unknown  elements  and 

not  0  nor  oo,  the  roots  of  the  equation  are  not  changed  thereby. 

Let  the  equation  p  =  q  be  any  equation,  and  n  any  number  not 

a  function  of  the  unknown  elements,  and  not  0  nor  oo  ; 

then  are  the  roots  of  the  equations  p  =  q,  n-p  =  n-q  identical. 

For,  write  the  equations  in  the  form  p  —  q  =  0,  n(p  —  q)  =  0  ; 

then    •.•  N  is  not  a  function  of  the  unknown  elements,  and  not 

0,  nor  00, 

.♦.  n(p  —  q)  vanishes  when  p  —  Q  vanishes,  and  then  only, 

and  conversely ; 
.*.  every  root  of  either  equation  is  a  root  of  the  other. 

Q.E.D. 

Note  1.  If  n  be  0,  n(p  —  q)  vanishes  for  any  values  of  the 
unknown  elements  that  make  p  —  Q  finite. 

If  N  be  X,   n(p  —  q)  may  not  vanish  when   p  —  Q   vanishes. 


1, 2,  §  4.]  GENERAL  PKOPErwTIES.  285 

If  N  be  a  function  of  the  unknown  elements,  n  may  vanish 
for  other  values  of  those  elements  than  those  values  that  make 
p  —  Q  vanish,  and  may  thus  cause  n(p  —  q)  to  vanish. 

If  N  be  a  function  of  the  unknown  elements,  n  may  become 
infinite  for  some  of  the  values  that  make  p  —  Q  vanish,  and 
n(p  —  q)  may  not  vanish. 

In  each  of  these  four  cases  the  equations  p— q  =  0,  n(p— q)=0 
may  not  have  all  their  roots  identical. 

E.g.,   let  X  be  any  entire  function  of  x,  and  a  any  constant ; 
then    •••  whatever  factors  x  has,  the  product   (x—a)'iL  has 
another  factor,  x  —  a. 
.',  whatever  roots  the  equation  x  =  0   has,  the  equation 
(a;  —  a)  •  X  =  0   has  also  the  root  a. 
So,       of  the  equation  a^  —  5x  -{-6    —0,  the  roots  are  2,3; 
but  of  the  equation  x^  —  5a^-\-  6x  =  0,  the  roots  are  0,  2,  3, 

I.e.,  by  multiplying  the   equation  by  a;  a  new  root,  0,  is 

introduced  which  does  not  satisfy  the  original  equa- 
tion  ar  —  5»-|-6  =  0,  and  is  not  a  root  of  it. 
So,       of  the  equation  x^  —  oxr -}- 6x^=0  the  roots  are  0,2,3; 
but  of  the  equation  ar— 5a;  +  6  =  0  the  roots  are  2,  3,  only, 

i.e.,  by  dividing  the  equation  b}'  x,  one  root,  0,  is  lost. 

So,  of  the  equation  3  —  a?  =  15  —  2  a;,  the  single  root  is  12  ; 
but  if  this  equation  be  multiplied  by   a;  — 1,    the  resulting 

equation   a^—  13a;  +  12  =  0,   has  two  roots,  12,1. 
So,       of  the  equation  a^  —  l=ax  —  a  the  roots  are  1 ,  a  —  1 ; 
and  if  both  members  be  multiplied  by  a; :  (a;  —  1 ) ,  the  re- 

sulting equation,  a;(a;-f- 1)  =  aa;,  still  has  the  root 
a;  =  a  —  1,  for  which  the  multiplier  becomes  neither 
00  nor  0, 

but  it  has  -{       •     a  o  ^^^  which  the  multiplier  becomes  ^  ^  * 

Note  2.  If  the  function  x  be  not  entire,  but  contain  a  term 
of  the  form  a  :  (x—a)  ;  then  x  may  become  infinite,  when  x  =  a, 
and  (x  —  a)  -x  may  take  the  form  0  •  oo,  which  may  or  may  not 
vanish  ;  and  multiplying  the  equation  x  =  0  by  a;  —  a  may  or 
may  not  introduce  a  new  root  into  the  equation. 


2S6  EQUATIONS.  ,  [XI.  tlis. 

E.g. ,    multiplicatiou  by  the  factors  a;,  a;  -f  1  generally  intro- 
duces the  roots  0,~1  ; 

1                       3 
but  if  the  equation    2  =  --fa;  —  1-| —    be  multiplied 

by  a;-(a;-f  1),  0,"1  are  not  introduced  as  new  roots. 

For     •••  X  contains  a  term  which  is  infinite  when  x  =  — 1 , 

a;-f-l 

and    •.*  this  infinite  term,  when  multiplied  by  the  zero,  ic  +  l, 

is  the  finite  number  3, 

.*.   (x-f  1)'X  does  not  vanish  when  x  =  ~\\  and  ~1  is  not 
a  root  of  the  new  equation. 

So,       X  •  X  does  not  vanish  when  a;  =  0  ;  and  0  is  not  a  root. 

But  if  the  equation   1 = —  6    be  multiplied  by 

aj"""!      x — x 

a;  —  1 ,  the  resulting  equation  is   a:^  —  7a;  -f-  G  =  0, 

whose  roots  ai-e  6,  1, 
whereof  6  satisfies  the  original  equation  and  is  a  root  of  it ; 
but     *.*  1  does  not  satisfy  it,  and  is  not  a  root  of  it, 

.'.  by  the  use  of  the  factor  a;  —  1  a  new  root  (a  stranger) 

has  been  introduced  into  the  equation. 

The  reason  is  manifest :  the  factor  x—  1  is  not  needed  to  clear 
the  equation  of  fractions  ;  for  if  the  terms  of  the  origi- 
nal equation  be  all  transposed  to  one  side  and  reduced 

1  —a? 

to  lowest  terms,  the  equation  becomes  7 =  0, 

\—x 

i.e.,  7— (l-ha;)=0,  whence  x=6  ;  and  there  is  no  other  root ; 

i.e.,  the  numerator  and  denominator  vanish  together  when 

a;  =  1 ,  and  the  value  of  the  fraction  0  :  0  is  2. 

So,       the  equation  1  -\ x-j-  — =  0  mav  be  cleared  of 

6  ar  —  1 

frattions  by  multiplying  by  Q{x—l)(x-^l),  and  be- 
comes 7a^  + 6a;— 13  =  0,  whose  roots  are  1,  — -V-; 

but     •.•   — -^  satisfies  the  original  equation,  and  1  does  not, 

.  • .  the  factor  a;  —  1  introduces  a  new  root  1 ,  but  a;  4- 1 
does  not  introduce  a  new  root. 


2, 3,  §  4.]  ^     GENERAL  PEOPERTIES.  287 

The  reader  may  search  out  the  reason  for  this  difference. 

So,       if  the  equation 1 =  0  be  multi- 

X  —  a      x-{-  a      x'  —  a" 

plied  by  all  its  denominators,  the  resulting  equation  is 

(2a;  —  1)  {xr  —  a-)  =  0,  whose  roots  are  ■^,  +  a,  —  a  ; 

but  if  it  be  multiplied  by  the  least  common  denominator, 

the  resulting  equation,  2  a;  — 1  =0,  has  a  single  root,  J. 

Of  these  three  roots  only  J  satisfies  the  original  equation. 

Theor.  3.    If  the  two  members  of  an  equation  he  raised  to  the 
same  integral  power ^  the  results  are  equal;  hut  it  is  possible  that 
the  new  equation  may  have  some  roots  not  found  in  the  old  one. 
For      if  p  =  Q,  wherein  p  or  q  or  both  of  them  are  functions 
of  some  unknown  element,  say  cc, 
then         p-  =Q^  p^  =  Q^  ...,  p"  =  Q**,  [II.  ax.  6 

p2_Q2  =  o,  p3-Q3=0,  ...,  P--Q"  =  0, 

i.e.,  (p_Q)(p-f-Q)  =  0,       (p_Q)(p2-|-PQ+Q2)  =  0,    ..., 

(P  -  Q)(p"-^  +  P"^2q  ...    _f_  QU-l)  ^  Q^ 

But      these  equations  are  satisfied  either  if  such  values  be 

given  the  unknown  that  p  —  q=0, 

or  that     p+Q  =  0,  p2+pq4-q2=  0,  •..p''-i-fp'»-2Q...  +  Q'^-i=  0; 

and  in  general  the  roots  of  the  equation  p  —  q  =  0  are  not 

the  same  as  the  roots  of  the  equations  p  +  q  =  0, 

p'  +  pQ  +  q'  =  0,  •••,    p'^-i  +  p'*"2q  _!-... +Q«-i=0. 

E.g.,  it  a;  =  5,    then   a;^=25,    and   a;  =  +5,~5; 

but         only  "^5    satisfies  the  original  equation  and  is  its  root. 

So,  if  1;/{0—x)  =  x—d,  then  a;^— 17a;+72  =  0,  and  a;  =  8,  9; 

but  9,.  not  8,  satisfies  the  original  equation,  and  is  its  root. 

AYere  that  equation  -^{d—x)  =  a;  —  9,  the  root  were  8,  not  9. 

Note.   Unless  the  reader  be  sure  that  every  step  he  has  taken 

.g  .  va  I  ,  ^^     ^^^^  ^^^j^  successive  transformed  equation 

'  reversible^ 

is  true^  whenever    ^^^^  previous  ones  are  true,  his  results  can 

'  only  when  ^ 

serve  merely  to  suggest  values  of  the  roots  for  trial.     If  any 

step  has  been^  jr^ev^sible  ^^^  P^'oblem  may  have^  ^^^^^  solu- 


288  EQUATIONS.  [XI.  tlis. 

tions  than  he  has  fouud.  In  particular,  he  must  have  multiplied 
by  uo  more  factors  containing  the  unknown  than  were  necessary 

to  clear  of  fractions,  and  must  have  w    ^        no  solutions  in 

takino;  like  ^  ^    -       of  both  members  ;  or  else  he  must  test  his 
^  '  roots  ' 

results  by  substituting  in  the  original  equation  or  equations,  and 
say ;  if  the}'  satisfy  the  equation  they  are  among  its  true  roots  ; 
if  not,  they  are  strangers  introduced  in  course  of  the  work.  The 
results  are  to  be  trusted  only  after  they  are  tested. 

Theor.  4.  If  all  the  terms  of  a  rational  integral  equation 
involving  one  unknoiun  element  be  transposed  to  one  side,  then : 

1 .  The  polynomial  so  formed  has  for  a  factor  the  excess  of  the 
unlcnoivn  element  over  any  root  of  the  equation; 

2.  Conversely,  if  this  iwlynomial  have  for  a  factor  the  excess 
of  the  unknown  element  over  any  given  number,  that  number  is  a 
root  of  the  equation. 

1.  Let  the  equation  x  =  0  be  any  equation  wherein  x  stands 

for  some  rational  integral  function  of  an  unknown  ele- 
ment X,  say  Aa;"+  Baf*"^4-caf*~^H [--rxt-^ sx-f-T, 

and  let  a  be  a  root  of  the  equation  x  =  0, 

then  is  x  measured  by  x—  a. 

For,  divide  x  by  x—a,  and  put  q,r  for  quotient  and  remainder  ; 
then   *. •  x  =  Q  •  (a;  —  a)  +  R,  for  every  value  of  x, 
wherein    r  is  independent  of  x  and  constant ; 
and     •.•  x=0   and   x  —  a=0  when  a;  =  a,  [%P- 

.-.  R  =  0; 
and     • .  •  the  division  of  x  by  a;  —  a  is  effected  without  remainder, 
.♦.  x  —  a  is  a  measure  of  x.  q.e.d. 

2.  Let  X  be  any  rational  integral  function  of  x,  and  let  a;  —  a 

be  a  measure  of  x  ; 
then  is  a  a  root  of  the  equation  x  =  0. 
For    • .  •  X  =  Q  •  (a;  —  a) ,  for  every  value  of  x,  and  there  is  no 

remainder,  [pJP' 

and     * . •  X  —  a  =  0   when  x  =  a, 

. *.  X  =  Q  •  0  =  0,    when  x  is  replaced  by  a, 
i.e.,  a  satisfies  the  equation  x=  0,  and  is  a  root  of  it.  q.e.d. 


3, 4,  §  4.]  GENERAL  PROPERTIES.  '  289 

CoE.  1.  Every  factor  of  x  that  is  itself  a  function  of  x  may 
be  put  equal  to  0,  and  the  roots  of  the  equations  so  formed  are 
roots  of  the  equation  x  =  0. 

Cor.  2.  No  rationalintegral  equation  x  =  0  has  more  roots 
than  the  function  x  has  linear  factors  [factors  of  the  form  x—a']\ 
and  if  the  equation  he  of  the  nth  degree,  it  has  not  more  than  n  roots. 
CoR.  3.  If  there  he  two  rational  functions  of  the  same  variable , 
neither  of  which  is  higher  than  the  nth  degree,  and  if  these  two 
functions  be  equal  for  more  than  n  finite  values  of  the  variable; 
then  are  the  two  functions  identical. 

Let  AX"  -\-  Bx''-'^  H h  so;  -h  T  =  a'x""  +  B'a;"-^  -\ f-  s'aj  +  t' 

be  a  true  equation  for  more  than  7i  finite  values  of  x  ; 
then  will  A  =  a',    b  =  b',   •••  s  =  s',    t  =  t', 

and  Aa;"  +  bx''~^-\ 1-  sx  +  t  =  A';t'"+  b'x''-'^-\ h  s'ic  +  t. 

For,  if  not,  the  equation 

(a— A')a:'*  -h  (b  -  b')  x''-'^-\ f-  (s  —  s')a;  +  (t-t')  =  0 

has  not  more  than  n  roots  ;  [cr.  2 

which  is  contrary  to  hypothesis  ; 

.-.    A  =  a',     B  =  b',   •••.  Q.E.D. 

Note.     The  roots  may  not  be  all  different. 
For  if  the  function  x  have  the  same  factor  used  two  or  more 
times,  then  the  equation    x  =  0    is  said  to  have  two 
or  more  equal  roots. 
In  general,  if  x  =  (a;  — a)^-  {x  —  hy-", 
wherein  p,  g,  •••  are  positive  integers  such  that  p+gH —  =  w; 
then         ot  is  a  p-fold  root,  b  a  q-fold  root,  and  so  on. 

E.g.,    ^  —  2>a3i?-{-^a^x  —  a^  =  {x  —  ay, 
and  the  three  roots  of  the  equation 

a?—  3  03I?  -\-  3  a^x  —  a^  =  0   are   a,  a,  a. 
So,       the  equation    (a; +  a)-(a;  — 6)^  =  0 

has  —a,  —a,  h,  b  for  its  four  roots. 
It  appears  later  that  a  set  of  equal  roots  are  the  limits  of  a  set 
of  unequal  roots,  and  that  if  the  equation  x  =  0  be  of  the  nWx 
degree,  it  has  n  roots,  equal  or  unequal,  real  or  imaginary. 


290  EQUATIONS.  [XI.  th.  5. 

Theor.  5.  i/",  of  the  rational  integral  equation  x=0,  the 
absolute  term  =  0,  some  root  »'  of  the  equation  =  0. 

Let  equation   x  =  0   be  written  t  =  —  s  a;  —  r  ar^ a  a;'*, 

and  let  s,  r,  •••  a  stand  fast,  while  a;,  t  vary,  and  t  =  0  ; 
then  will  some  root  a;'  =  0. 

For     •.•  D„T,  =-s-2Ra;' ,  [VII.  ths.  13,17 

=  —  s,    a  finite  number  ;  [a;'  =  0 

.•.the  ratio  incT :  inc  a'   is  finite  ;   and  the  two  infinitesi- 
mals are  of  the  same  order. 
But     •.*  T  =  0  when  a;'=0, 

.*.  if  T  ~  0  be  small,  so  is  a;'  '^  0  ; 
2.6.,  a;'  =  0  when  t  =  0.  q.e.d. 

CoE.  1.  If  tJie  absolute  term  be  0,  then  is  0  a  root  of  the  equation. 
For,  if  0  be  put  for  x,  the  equation  x  =  0  is  satisfied,    q.e.d. 

Cor.  2.  7/*  a,  the  coefficient  of  the  highest  power  of  the  unknown 
element  in  x,  =  0,  then  a  root  of  the  equation  =  oo. 

For,  if  X  be  replaced  by  y~^  in  the  equation   x  =  0, 
that  equation  takes  the  form 

yn         yn    1        yn    2  y^         y 

whence    Ty""  -\-  st/**"^  +  iiy''~^-\ h  c?/^  +  b?/  -f-  a = 0  ;  [mult,  by  ?/" 

and  if  the  absolute  term  a  =  0, 

then         some  root  2/'=  0,  and  some  root  a;',  =  1 : 2/'==  Qc.  Q.e.d. 

Note.  If  the  last  two,  three,  •••  of  the  coefficients  •••,  r,  s,  t 
be  zero,  or  approach  zero,  so  do  as  many  of  the  roots ;  and  if 
the  first  two,  three,  •••of  the  coefficients  .a,  b,  c,  •••  approach 
zero,  as  many  of  the  roots  approach  infinit}*. 

E.g.^  if  A,  B,  c,  R,  s  be  infinitesimals  of  the  first  order,  t  be 
zero,  and  d,  q  be  finite, 

then         three  of  the  roots  are  infinites,  each  of  the  order  ■^,  two 
are  infinitesimals,  each  of  the  order  ^,  and  one  is  zero. 


pr.l,  §  6.]  SniPLE  EQUATIONS  INVOLVING  ONE  UNKNOWN.  291 

§  5.     SIMPLE    EQUATIONS    INVOLVING    ONE    UNKNOWN. 

PrOB.  1.  To  SOLVE  A  SIMPLE  EQUATION  INVOLVING  ONE 
UNKNOWN   ELEMENT. 

Multiply  both  members  of  the  equation  by  the  I.  c.  mlt.  of  the 
denominator's,  if  any.  [th.  2 

Transpose  to  one  member  all  terms  that  involve  the  unknown 
element^  arid  to  the  other  member  all  other  terms.  [th.  1 

Reduce  both  members  to  their  simplest  form,  exhibiting  or  can- 
celling any  common  factors. 

Divide  both  members  by  the  coefficient  of  the  unknown  ele- 
ment. [II.  ax.  5 

To  test  the  work,  replace  the  unknown  element  by  the  result  so 
found,  in  the  original  equation. 

E.g.,    if  J  (a; +12)  =  ^(G-f  3a;)- !«,  [a;  unk. 

then    •.•  7a; +  84  =  36  + 18a; -7a;,  [mult,  by 42 

and           (7  -  18  +  7)  a;  =  36  -  84,  [trans.  84, 18a;,  —  7a; 

i.e.,          —4a;  =—48,    and  a;  =12;  [div.  by  —  4 

and     •.•  1(12 +  12)  =  1(6 +  36) -2,  [repl.a;byl2 
.-.  12  is  the  root  sought. 

So,       if(2^i±i^\.  +  _^^  =  3c.  +  ^a;-^, 
a{a-^bY  {a^bf  a        a  +  & 

then   •.•   (2a  +  6)62(a  +  6)a;  +  a362 

=  3  ac  (a  +  6)2  a;  +  6  (a  ^bfx-  Sa^bc  (a  +  6)^ 
.-.  a^b--\-Sarbc{a-^by 

=  (3 ac  +  6)  (a  +  b^x  -  (2a  +  b)  b\a  +  b)x, 
i.e.,  a'b  [a6  +  3  c(a  +  5)^]  =  a  {a  +  6)  [a&  +  3  c(a  +  bf]  x, 

X  =      a" b  lab -\- 3  c  (a  +  by]      ^    ah 
~  a{a  +  b)  [ab  +  3  c{a  +  6)-]  ~  a  +  6 

This  value  satisfies  the  given  equation,  and  is  the  root  sought. 
Note  1.  A  simple  equation  can  have  but  one  root. 
For  any  such  equation  may  take  the  form     ax  —  b  —  0,     one 
linear  factor.  [th.  4  cr.  2 


292  EQUATIONS.  [XI.  tlis. 

Note  2.  Equations  not  simple  sometimes  reduce  to  simple 
equations,  and  may  be  solved  like  them. 

E.g.,  if  v^- V[«-V(i-^')]=i; 

then   •••  ^[x  —  ^{l  —  x)~\  —  -^x  —  \^  [trans. ^^o;,  change  signs 

a;  — -^/(l  —  x)  =  X  —2y/' X  -\- 1,  [sqr.  both  mem. 

^(1  —  x)  =  2^x—\,  [cancel  .T,  change  signs 

1  —  a;  =  4a;  — 4-y/x+ 1,  [sqr.  both  mem. 

.  4:^x  =  ox,  [trans. 

.  16a;    =25ar^,  [sqr.  both  mem. 

a;(16  —  25a;)  =  0,  [trans.,  factor 
it-  =  0    or    =  ||. 

Both  of  these  results  satisfy  the  given  equation,  and  are  roots. 

For       Vif-V[M-V(l~i>f)]=l    if   the  second  radical 
be  negative,  and  the  other  two  positive  ; 

and  V0-V[0-V(l-0)]  =  1   if  the  last  two  radicals 

take  their  negative  values. 

But  if  the  signs  of  the  radicals  be  restricted,  the  equation  ma}^ 
have  no  solution. 

Eg-,   V«-V[»-V(i-^)]=i. 

VaJ-V[«-V(l-^)]=l- 

Note  3.  General  Discussion:  Every  simple  equation  in- 
volving one  unknown  may  be  reduced  to  the  form 

ax-\-b  =  a'x-{-b\    whose  general  solution  gives 
x  =  (b'  —  b):  (a  — a')  ;    and  there  are  three  cases  : 

(a)  a=^a' ;  then  x  has  a  single  value,  positive,  negative, 
or  zero,  that  satisfies  the  equation. 

(6)    a  =  a\    b=^b';    then   x  =  cc. 

This  result  may  be  interpreted  in  the  language  of  limits  by 
saying  that  if  a,  a'  be  variables,  or  either  of  them,  and  if 
a=^a'  but  a  =  a',  then  x  grows  larger  and  larger  without  bounds. 

E.g.,  if  A,  a'  travel  along  the  same  road  in  the  same  direc- 
tion at  a,  a'  miles  an  hour,  and  if  a'  be  {b'—b)  miles  ahead  of  a. 


6,6,  §G.]      -  ELIMINATION.  293 

then  the  quotieDt  {b'  —  b)  :  (a  —  a')  is  the  time  before  they  will 
be  together. 

If  the  hour! 3'  gain,  a  —  a',  be  small,  that  time  is  long ;  if 
there  be  no  gain,  i.e.,  if  a=:a\  they  will  never  be  together, 
and  there  is  no  value  of  x  that  satisfies  the  equation. 

(c)  a  —  a',  b  —  b'-,  then  x  =  0:  0,  and  the  equation  is  satisfied 
by  any  number  whatever. 

In  the  example  above  (6) ,  a,  a'  are  now  together  and  they 
will  always  be  together. 

§  6.    ELIMINATION. 

Theor.  6.  If  there  he  two  or  more  unTcnown  elements  and  a 
system  of  two  or  more  independent  simultaneous  equations  that 
involve  them,  then  the  roots  are  not  changed  thereby  if  any  one  of 
these  equations  be  replaced  by  the  sum  of  this  equation  and  any 
other  or  others  of  them. 

Let  the  equations  p  =  Q,   p'=  q',   p"  =  q",  ...  be  any  system 
of  simultaneous  equations,  and  for  the  equation   p  =  Q 
put  the  equation   p  -j-  p'  =  q  +  q', 
or  p-f  p'  +  p"  =  q4-q'  +  q",    or-..;  [II.  ax.  2 

then  will  the  roots  of  the  system  of  equations, 

p  +  p'  +  ---=Q  +  Q'+---,    p'  =  q',    p"  =  q",   -, 
be  identical  with  the  roots  of  the  system  first  given. 

For     *.*  when   p  =  Q,   p'  =  q',    p"  =  q",  .••,   then  also 

p  +  p'+p"+---  =  q  +  q'+q"4--"; 

.*.  whatever  set  of  values  satisfy  the  equations 

p  =  Q,   p'=q',    p"=q",  ...,   the  same  values  satisfy 

the  equation   p4-p'4-p"H =Q  +  q'  +  Q"H ; 

and  conversely.  q.e.d. 

Cor.  In  such  a  system  of  equations  the  roots  are  not  changed 
if  before  the  addition  one  or  more  of  the  equations  be  multiplied 
by  any  factor  not  a  function  of  the  unknotvn  elements,  and  not  0. 


294  EQUATIONS.  [XI.  th. 

Theor.  7.   If  one  equation  of  a  system  be  solved  for  any  one  un- 
known element^  in  terms  of  the  other  unJcnown  elements  that  enter 
into  it,  then  the  mots  of  the  system  are  not  changed  thereby,  if  in 
the  other  equations  this  element  be  replaced  by  the  value  so  found. 
For,  let  p  =  Q,  p'  =  q',  p"  =  (;>",  •••  be  a  sj'stem  of  equations 
involving  x,  y^z,  •••  in  any  way;  solve  the  equation 
p  =  Q  for  X,  giving  x=f{y,  z,  •••),  and  substitute  this 
expression  for  x  in  the  other  equations,  giving  them 
the  new  forms  r'  =  s',  r"  =  s",  •••  ; 
then   •••  X  ancl/(y,  2,  •••)  have  the  same  values, 

.-.  whatever  values  of  x,  y,  z,  •••  make  identities  of  the 
equations  p=q,  p'=q',  p"=q",  ...,the  same  values 
make  identities  of  the  equations  r'=  s',  r"=s",  •••, 
and,  conversely,  whatever  values  of  y,  z,  »-•  make  identities  of 
the  equations  r'=s',  r"=s",  •••,  the  same  values  of 
y,  2,  •••  make  identities  of  the  equations 
p  =  Q,   p'  =  q',   p"  =  q",  •.., 
I.e.,  both  systems  have  the  same  roots.  q.e.d. 

Note.  By  aid  of  Theors.  1,  2,  3,  6  a  system  of  n  independent 
simultaneous  equations  containing  n  unknown  elements  ma}'  be 
reduced  to  a  new  system  of  w  —  1  equations,  containing  n  —  l 
unknown  elements,  whose  roots  are  identical  with  the  roots  of 
the  original  system,  and  these  n—l  equations  to  ?i~2  equa- 
tions, •  ••,  to  two  equations,  to  one  equation. 

The  process  by  which,  one  after  another,  the  several  un- 
knowns are  removed  from  the  system  of  equations  is  a  case  of 
elimination.  In  general  when  from  two  or  more  (say  n)  given 
equations  a  new  equation  is  got  that  is  free  from  at  least  n  —  l 
of  their  elements,  those  elements  are  eliminated  between  the  given 
equations;  and  the  new  equation,  or  its  first  member  when  the 
second  member  is  zero,  is  the  resultant  of  the  given  equations. 

The  elimination  is  reversible  when,  whichever  ti  —  1  of  the 
given  equations,  together  with  the  resultant,  were  known  to  be 
true,  the  remaining  equation  would  necessarily  be  likewise  true  ; 
otherwise,  the  elimination  is  irreversible. 


7,  §  6.]  ELIMINATIOiT.  295 

PrOB.    2.     To    ELIMINATE    AX   UNKNOWN    ELEMENT  FROM   A  PAIR 
OF   EQUATIONS    INVOLVING   THE    SAME   TWO   UNKNOWN   ELEMENTS. 

(a)  Simple  equations. 

FIRST    METHOD,  ADDITION   AND    SUBTRACTION. 

Find  the  least  common  multiple  of  the  coefficients  of  that  element 
which  is  to  be  eliminated;  divide  it,  in  turn,  by  these  coefficients, 
and  multiply  the  two  equations  through  by  their  quotients. 
Subtract  one  equation  from  the  other,  member  from  member. 
E.g.,  to  elimioate  x  from  the  pair  of  equations 
6x-{-7y  =  85,    2x+Sy  =  SS'. 
then    •••  the  l.c.mlt.  of  the  coefficients  2,  6  is  6, 

...  6a;4-7?/  =  85,    6a;-f9?/  =  99,  [mult.byl,3 

2?/=  14.  [subtract 

So,  to  eliminate  x  from  the  pair  of  equations 
Laj  +  M  =  0,    l'cc4-m'=0, 
wherein   l,  m,  l',  m',  are  any  expressions  that  do  not  contain  x, 

but  which  may  contain  other  unknown  elements  : 
then   • .  •  LL'ic  +  l'm  =  0,    ll'cc  -f  lm'  =  0,  [mult,  by  l',  l 

.-.  lm'— l'm    =0.  [subtract 

Note.  The  work  is  often  best  arranged  as  follows  : 
Wonte  the  given  equations  under  each  other,  and  at  the  right, 
their  respective  multipliers  ivith  such  signs  that  the  new  equation 
may  be  the  algebraic  sum  of  the  products  of  the  given  equations 
by  their  midtipliers.  If  there  be  two  columns  of  multipliers,  one 
to  eliminate  each  unknoivn,  write  first  the  column  to  be  first  used. 
When  small,  the  partial  products  can  be  obtained  and  added 
mentally,  and  only  the  sums  written  down.  Detached  coeffi- 
cients can  be  used  in  part  of  the  work. 

E.g.,  the  first  of  the  above  examples  becomes  : 


6a;  +  72/  =  85 

3 

-1 

ex-i-7y  =  85 

2a;  +  3?/  =  33 

~7 

3 

or 

2    3    33 

4a;     =255 

=  24 

4       24 

-231 

2    14 

2?/  =  14 

x=Q,  y=7 

x=.Q,  y=7. 

[3-1 
L-7  3 


296  EQUATIONS.  [XI.  pr. 

SECOND    3IETHOD,    COMPARISON. 

Solve  both  equations  for  that  element  wJiich  is  to  be  eliminated. 
Put  the  two  values  thus  found  equal  to  each  other. 
E.g.,  to  eliminate  x  from  the  pair  of  equations 
6a; +  72/ =  85,    2a; +  3?/ =  33: 
then         a;  =  |(85— 72/)  =  i(33  — 32/),  [sol.  both  eq.  for  a? 

THIRD   METHOD,   SUBSTITUTION. 

Solve  either  equation  for  that  element  which  is  to  be  eliminated. 
In  the  other  equation  replace  this  element  by  the  value  so  found. 
E.g.,  to  eliminate  x  from  the  pair  of  equations 
6a;-f-72/  =  85,    2a;  +  32/  =  33: 
then    •.•  a;  =  ^(33  — 32/),  [sol.  2d  eq.  for  a; 

.-.  99  —  9?/4-7y  =  85.  [repl. a;in  Isteq. 

(6)  Equations  of  degree  higher  than  the  first. 
Of  the  three  methods  of  elimination  shown  above  (a)  some- 
times one,  and  sometimes  another  is  most  available. 

In  the  pair  of  equations  I  =  a?-**  \   s  =^ [to  elim.  n 

r  —  1 

the  method  of  substitution  is  best :   multiply  the  first 
equation  by  r,  and  replace  ar""  b}'  Ir  in  the  other  ; 

i.v                    Ir  —  « 
then         s  — 

r  —  1 
In  the  same  pair  of  equations  [to  elim.  a 

the  method  of  comparison  is  best:    solve  both  equa- 
tions for  a,  and  put  the  values  equal ; 
then         -l.  =  s(r-l)_ 

In  the  pair  of  equations    x^-\-y  =  ll,    y^-\-x  —  l        [to  elim.  a; 
the  first  method  is  less  easy  ;  but  the  other  two  are  available. 

2.  •.•  ar=ll-2/,     3;^  =  (7-2/^)2  =  49  -  14  2/2  +  2/S 

.-.  11  — 2/  =  49  —  142/^-f  2/^  [comparison 

3.  •.•  X  ={l-f~), 

.'.  x'-\-y  =  ll  gives  4,9 —14: y--\-y^-\-y  =  11.  [substitution 


2,  §  6.]  ELIMINATION.  297 

FOURTH   METHOD,    DIVISION. 

Reduce  the  equations  to  the  form  p  =  0,  Q  =  0,  ivherein  p,  q 
a?'e  functions  of  x,  y. 

Divide  f  by  q,  q  by  the  remainder^  and  so  on,  as  in  finding  the 
h.  c.  msr.  of  two  entire  numbers,  until  some  remainder  is  found 
that  is  free  from  the  element  to  be  eliminated,  or  that  has  a  com- 
mon measure  ivith  p,  q,  and  the  successive  remainders. 

If  this  remainder  do  not  contain  such  a  common  measure, 
equate  it  to  0  for  the  resultant  sought. 

If  a  common  measure  ofp,  q  be  found,  divide  each  of  them,  or 
any  two  successive  remainders,  by  this  measure,  and  with  the  quo- 
tients proceed  as  before  to  find  a  resultant. 

If  the  solution  of  the  given  equations  be  sought,  then : 

Solve  the  resultant  for  the  unknown  element  involved  in  it;  re- 
place this  element  in  the  next  previous  remainder  by  the  values  thus 
found;  equate  to  0,  and  solve  for  the  other  unknown  element. 

Equate  to  0  the  common  measure,  if  any,  o/  p,  q  ;  if  the  new 
equation  thus  found  involve  but  one  unknown  element,  solve  it 
therefor;  but  if  it  involve  both  unknown  elements,  give  to  either 
of  them  any  value  whatever,  and  solve  for  the  other. 

E.g.,  to  eliminate  y  from  equations   p  =  0,    Q=  0,    wherein 
TisX'y*—2x-\-l  -y^—x^—x^—x—l  •y^-\-a?—x^—x-\-l-y-{-2, 
and  Q  \sX''f-\-s(?  —  2a;— 1  - y"^  —  x^ -\- x —  I  -y  —  2'X^—x—l  : 

Divide  p  by  q  ;  the  remainder,  •R,  =  x-y^  —  2x-\-l'y-\-2; 
so,  divide  q  by  r  ;   the  remainder,  s,  =  oi^ —l-y —  2 ', 
then   •••  R,  s  have  the  h.  c.  msr.  y  —  2, 

and  the  quotients  are  xy—1,  a^—l, 
.'.  the  resultant  is   a;^— 1  =  0,    whose  roots  are  '''1, 
and  equations  p=0,  Q=0  are  satisfied  when,  and  only  when, 

either  a;  =  +1, +1-2/— 1  =  0;  or  fl;=-l,  "1 -2/— 1=  0  ; 
or  a?  =  any  value,  y  =  2. 

So,  to  apply  the  fourth  method  to  the  pair  of  equations, 

3a^-4xy-{-5y^-Gx-^7y=120,    2i^-Sxy+5y'=10S: 

Write  p,  Q  in  the  form    lx^H-  mx  +  n,     j.'x^-\-  m'x  +  n', 
wherein   l,  m,  n,  l',  m',  n',  may  contain  y  but  not  x. 


298  EQUATIONS.  [XI.  prs. 

§7.     SIMPLE    EQUATIONS,    TWO   OR   MORE   UNKNOWNS. 

PrOB.  3.  To  SOLVE  A  PAIR  OF  SIMPLE  EQUATIONS,  WHEREOF 
ONE   HAS   TWO   UNKNOWN  ELEMENTS,   AND  THE   OTHER  BUT  ONE. 

Solve  that  equation  which  has  hut  one  unknown  element,   [pr.l 

Replace  this  element  by  its  value  in  the  second  equation,  and 

solve  for  the  other  unknown  element.  [th.  4  cr.  2 

E.g.,  to  find  x,  y  from  the  pair  of  equations 
6a;+7y  =  85,     4a;  =  24: 
then         x=Q>,     36 +  7?/ =  85,     y=l. 

PrOB.  4.  To  SOLVE  A  PAIR  OF  SIMPLE  EQUATIONS,  WHEREOF 
BOTH   HAVE   THE    SAME   TWO   UNKNOWN    ELEMENTS. 

Combine  the  two  equations  so  as  to  eliminate  one  unknown  ele- 
ment, and  form  an  equation  involving  the  other  unknown  element. 

Solve  this  equation  for  its  unknown  element,  replace  this  element 
by  its  value  in  either  of  the  given  equations,  and  solve  the  equation 
so  found  for  the  other  unknown  element. 

For  a  check,  replace  the  two  unknown  elements  by  their  values 
in  either  of  the  original  equations. 

E.g.,    to  find  x,  y  from  the  pair  of  equations 
Qx-\-ly  =  m,     2a;  +  32/  =  33; 
then   •.•  ■f(85-6a;)  =  J(33-2a;),  [elim.2/ 

.-.  255 -18a;  =231  — 14a;, 
.-.   — 4a;=— 24,     a;  =  6; 
.-.  36  +  72/  =  85,     y^l. 
So,  to  find  X,  y  from  the  pair  of  equations 
ax  -\-by  =  c,      a'x  +  b'y  =  c' : 
then   '.'  o:b'x-^bb'y  =  cb',-    ba'x -\-bb'y=  be',  [II.  ax.  4 

.-.   {ab'-a'b)x={cb'-c'b),   a;=^^'~^'^, 

ab'  —  a'b 

cft^-c^ft  ac'-g'c. 

ab'-a'b       •'      '  ^     ab'-a'b 


3, 4,  §  7.]  SIMPLE  EQUATIONS,  TWO  OR  MORE  UNKNOWNS.  299 

Note  1.  The  values  of  the  two  unknown  elements  may  be 
got  independently  of  each  other,  by  separate  eliminations ;  or 
else,  having  found  one  of  them,  the  other  may  be  written  by 
symmetry. 

E.g. ,    if  ax -^  by  =  c,    a^x  +  Vy  =  c',  [above 

then    *.•  these  equations  are  not  altered  by  interchanging 
a  with  6,    a'  with  h\    and  x  with  ?/, 
.*.  a;  is  the  same  function  of  a,  6,  a',  6'  as  is?/ of  5,a,&',a'; 

.*.  the  value  of  either  x  ovy  \q  found  from  that  of  the  other 
by  interchanging  a  with  h  and  a'  with  h\ 

So,  if  a;  +  2/ =  a,    x  —  y—hi 
then  {  ^  is  the  half  ^  ^Xrence  ^^  ^'  ^' 

Note  2.  Incompatible  equations  :  If  two  given  equations 
be  incompatible,  no  solution  is  possible. 

J57.gr. ,  the  equations    2a;  +  32/=13,    2a;  +  3?/=15 
are  incompatible ; 
for  their  resultant,    0  =  2,    is  absurd. 

Note  3.  Dependent  equations  :  If  one  equation  be  depen- 
dent on  the  other,  and  derivable  from  it,  there  is  no  single  solu- 
tion, but  an  infinite  number  of  solutions. 

E.g.^  the  equations  2aj  +  32/  =  13,  6ic-f-92/  =  39  are  but 
one  equation  in  two  forms,  and  any  value  may  be 
given  to  either  of  the  unknown  elements,  and  the 
corresponding  value  of  the  other  computed. 

Note 4.  General  formula:  The  two  equations  ax-^by=c, 
a'x  -\-b'y  =  c'  are  the  type-forms  of  every  pair  of  two-unknown 
first-degree  equations  ;  their  solution  gives  : 

x=(cb'-c'b):(ab'-a'b),   y  =  (aG' -a'c)  :  (ab'- a'b). 

The  solution  of  this  pair  of  equations  embraces  the  solution  <^^ 
of  all  such  pairs  of  equations :    the  reader  may  translate  the   ^^'"'^^ 
formulae  into  a  practical  rule  for  such  solutions  without  the 
intermediate  steps. 


300  EQUATIONS.  [XI.  pr. 

Note  5.  General  discussion  :  There  are  three  general  cases, 
(a)  ah'=^a'b] 
then         a,  y  have  single  values,  positive,  negative,  or  zero,  that 
satisfy  both  the  equations. 
(6)  ab'  =  a'b,   cb'^c'b; 


then 

a;  =  oo,     y  =  co. 

For 

•.'  ab'    ==a'b,      cb'   ^c'b, 

[iiyp- 

.-.  a:a'  =  b:b\    c:c'=^b:b', 

[II.  ax.  13 

.-.  a:a'=^c:c', 

[II.  ax.  8 

and 

ac'     =^  a'c. 

Q.E.D.     [II.  ax.  13 

And 

'.'  a6'-a'6  =  0,     cb'-c'b^O, 

ac' 

-a'c^O, 

.-.  »,  =(c6'-c'6)  :  (ab'-a'b), 

=  00 

and  y,  =(ac'—a'c)  :  (ab'—a'b),  =  oo.  q.e.d. 

This  result  may  be  interpreted,  in  the  language  of  limits,  by 
saying  that  if  a,  a',  6,  6',  be  variables,  or  either  of  them,  and  if 
db'  =^  a'b  but  ab'  =  a'b,  then  ic,  y  grow  larger  without  bounds. 
E.g.,  if  ax-^by=c,   a'x  +  b'y  =  c'  be  the  equations  of  two 
straight  lines, 
then         the  values  of  a;,  y  that  satisfy  both  equations  are  the 
co-ordinates  of  the  meeting-point  of  the  two  lines. 
If  ab'  =  a'b,  then  a:a'  =  b:b',  the  two  lines  approach  paral- 
lelism, the  point  of  intersection  recedes  to  a  great 
distance,  and  the  values  of  x,  y  become  very  great. 
If  ab'  =  a'b,  then  a :  a'  =  6  :  5',  the  two  lines  are  parallel,  they 
have  no  meeting-point,  and  there  are  no  values  of 
X,  y  that  satisfy  both  equations, 
(c)  ab'=a'b,    cb'=c'b; 
then         a;  =  0:0,     y  =  0:0, 

and  the  equations  are  equivalent,  and  satisfied  b}"  giving 

any  value  to  one  unknown  element  and  computing 
the  corresponding  value  for  the  other. 
For     •••  ab'    =a'b,      cb'    —c'b,  [liyp* 

.'.  a:a'  =  b:b',   c:c'  =  b:b', 


a:a'  =  c:c', 


Q.E.D. 


4,  §  7.]     SIMPLE  EQUATIONS,  TWO  OE  MORE  UNKNOWNS.  301 

In  the  above  example  (h)  the  two  lines,  under  the  special 
conditions  given  in  (c) ,  are  coincident,  and  every  point  is  a 
common  point.  In  general  any  value  may  be  assumed  at  ran- 
dom for  one  co-ordinate,  and  the  other  may  then  be  computed. 

If   ab'  =  a'b,  there  also  appear  the  following  special  cases  : 

(d)  a'  =  0,    6'  =  0,   c'=5fcO; 

then         ic  =  —  6c' :  0,   y  =  ac' :  0,   which  values  are  infinite. 

(e)  a'  =  0,    6'  =  0,   c'  =  0; 

then         a;  =  0:0,   2/  =  0:0,  which  values  are  indeterminate. 

(/)  6  =  0,   6'  =  0,   OAi'^a'c, 
then         a;  =  0  : 0,   y  —  (ac'  —  a'c)  :  0,  of  which  values  one  is  in- 
determinate and  the  other  infinite. 

(g)  6  =  0,   6'  =  0,   ac'=a'c; 
then         the  equations  are  equivalent, 
and  a;  =  0  :  0  =  c  :  a  =  c' :  a', 

and  2/  =  0  :  0,    and  is  indeterminate. 

(h)  a  =  0,   6  =  0,    6'=0,   a'r^O,   c:?i=0; 
then         a;  =  0  :  0,   ?/  =  —  a'c :  0,  of  which  values  one  is  indeter- 
minate and  the  other  infinite. 

(i)  a  =  0,   6  =  0,   6'  =  0,   c=0; 
then         x  =  c':a\   y  =  0  :  0,  of  which  values  one  is  determinate 
and  the  other  indeterminate. 
(j)  a  =  0,    a'  =  0,    6  =  0,    6'  =  0; 
then         a;=  0  :  0,    ?/  =  0  :  0,    which  values  are  indeterminate  ; 
but  not  both  are  finite  unless   c  =  0,  c'  =  0. 

The  reader  may  interpret  these  results,  and  illustrate  them 
by  the  meeting,  when  possible,  of  two  straight  lines. 

Two  important  special  cases  appear  when   c,  c'  both  vanish. 

(k)  ah'=^a'b,   c  =  0,    c'=0; 
then         a;  =  0,   y  —  0. 

(l)  ab'  =  a'b,   c=0,   c'  =  0; 
then         a;  =  0  :  0,   2/  =  0  :  0,   which  values  are  indeterminate. 

The  reader  may  interpret  these  results,  and  illustrate  them  by 
the  meeting  of  two  straight  lines :  he  will  observe  that  in  both 
cases  the  two  lines  pass  through  the  origin ;  in  the  first  they 
meet  there  ;  in  the  second  they  coincide  throughout. 


302  EQUATIONS.  [XI.  pr. 

PrOB.    5.      To    SOLVE    A    SYSTEM    OF    71    INDEPENDENT    SIMPLE 
EQUATIONS   THAT   INVOLVE   THE    SAME    U   UNKNOWN   ELEMENTS. 

Combine  the  n  equations,  two  and  two,  in  n  —  1  tvays,  so  that 

each  equation  is  used  at  least  once,  and  so  as  to  eliminate  the 

same  unknown  element  at  each  operation;  thereby  form  n— 1 

equations  involving  the  same  n  —  1  unknown  elements. 

So,  combine  these  d  —  1  equations,  and  thereby  form  n  —  2 

equations  involving  the  same  n  —  2  unknown  elements;  and  so  on 

till  there  results  one  equation  involving  but  one  unknown  element. 

Solve  this  equation,  and  replace  the  unknown  element  by  its 

valu£  in  one  of  the  two  equations  involving  tivo  unknown  elements. 

Solve  this  equation  for  the  second  unknown  element,  and  replace 

these  two  elements  by  their  values  in  one  of  the  three  equations 

involving  three  unknown  elements;  and  so  on. 

E.g.^  to  find  x,  y,  z  from  the  system  of  equations 

a;-f-22/+32=14,   3a;+2?/+z=10,    6x-f9?/+13«=63 : 

then         2a;— 22=— 4,   15a;— 17z=  — 36,[elim.2/fr.eqs.l,2;2,3 

.*.  4a;  =4,  [elim.  2 

.-.  a;=l,    y—2,    2  =  3.  \yQ^\.x,y 

So,       to  find  X,  y,  z,   from  the  system  of  equations 

ax+by+cz=d,  a'x-\-b'y-\-c'z=:d',  a"x-\-b"y+c"z=d": 

d  —  by  —  czd'  —  b'y  —  c'z     d"  —  b"y  —  c"z 

then   •.•  x= ^ = ^ = f- , 

a  a'  a" 

.'.  a'd^a'by —  a'cz  =  ad' —  ab'y —  ac'z 

and  a"d'  -  a"b'y  -  a"c'z  =  a'd"  -  a'b"y  -  a'c"z,       [elim.  x 

.       ^  ad'-a'd-\-  (a'c-ac')z_a'd"-a"d'-^  (a"c'-a'c")  z 

'  '  ^  ab'-  a'b  a'b"  -  a"b' 

.-.   (a'b"  -  a%')  {ad'  -  a'd)  -f  (a'b"  -  a"b')  {a'c  -  ac')z 

=  (ab'-a'b)  {a'd"-a"d')  +  (ab'-a'b)  (a"c'-a'c")z, 

^  ^  jab' -a'b)  (a'd"-a"d')  -  (a'b"-a"b')  (ad' -a'd) 

(ab'-a'b)  (a'c"  -  a"c')  -  (a'b"-a"b')  (ac'  -  a'c) 

^  ab'd"  +  a'b"d  +  a"bd'  -  a"b'd  -  a'bd"  -  ab"d' 

~  ab'c"  +  a'b"c  +  a"bc'  -  a"b'c  -  a'bc"  -  ab"c' ' 

The  reader  may  write  the  values  of  x,  y  by  symmetry. 


6,  §  7.]     SIMPLE  EQUATIONS,  TWO  OP.  MORE  UNKNOWNS.  303 

Note  1.  The  equations  must  be  so  combined  that  no  m  of 
the  n  —  1  equations  got  by  the  elimination  of  one  unknown  ele- 
ment shall  represent  less  than  m  -f-l  of  the  original  n  equations  ; 
and  that  no  m  of  the  n  —  2  equations  got  by  the  elimination  of 
two  unknown  elements  shall  represent  less  than  m  4- 1  of  those 
n  —  1  equations  ;  and  so  on.  Otherwise  the  m  —  1  equations, 
or  the  m  — 2  equations,  •••,  will  not  be  independent,  and  no 
determinate  solution  will  finally  be  got. 

Note  2.  All  the  unknown  elements  not  involved  in  every 
EQUATION.  An  unknown  element  that  does  not  appear  in  sluj 
equation  may  be  considered  as  already  eliminated  from  it,  and 
the  work  is  shortened  by  so  much.  Those  unknown  elements 
that  appear  in  the  fewest  equations  may  be  eliminated  first. 

E.g.,  to  find  07,  y,  z,  t,  u  from  the  system  of  equations 

^x—2z-\-    w  =  41,  (1) 

7y-5z-    t  =12,  (2) 

Ay  —  ^x-{-2u=    5,  (3) 

3?/  — 4i<4-3f  =    7,  (4) 

7z-5u=n:  (5) 

Of  these  equations,  x  appears  in  two,  y  in  three,  z  in  three, 
u  in  four,    t  in  two. 

Equations  1,  3  may  be  combined  to  eliminate  x,  and  equa- 
tions 2,  4  to  eliminate  t,  and  there  result  two  new 
equations  involving  y,  z,  u. 

These  two  equations  may  be  combined  to  eliminate  y.,  and 
there  resalts  one  equation  involving  2;,  u. 

This  last  equation  may  be  combined  with  equation  5  to  elimi- 
nate either  2;  or  w  at  pleasure. 

Note  3.  Particular  artifices  :  The  equations  may  have  a 
symmetry,  as  to  the  unknown  elements  or  functions  of  them, 
that  permits  shorter  processes  than  those  of  the  general  rule. 
Sometimes  the  sum  of  the  unknown  elements,  or  of  the  func- 
tions of  them,  may  be  got  first. 


304  EQUATIONS.  [XI.  pr. 

E.g.^  to  find  a;,  y,  2,  ^  v  from  the  system  of  equations 
y  +  z-\-  t->rv  =  a,  (1) 

z +^ +'?^  +  a;  =  6,  (2) 

t  Jtv  +  x  +  y  =  c,  (3) 

^+a;  +  2/  +  2;=cZ,  (4) 

«  +  2/+2  +  ^=e;  (5) 

then   •••  4a;  +  42/^-4z4-4i  +  4^;  =  a4-5  + c  +  d  +  e,  [add 

and  ic  =  i(  — 3a4-6  4-c  +  <^ +e)»  [sub.eq.l 

2/  =  :|^(a  — 364-c  +  cZH-e),    and  so  on. 
So,  to  find  a?,  y,  2  from  the  system  of  equations 
1  ,  1_  4       1  ,  1_11       1      1_1. 
xy      10      yz      60      zx4: 

then  •.•  ?H-?  +  -  =  ^,     i  +  i  +  l  =  ^,  [add,div.by2 

*'*  a;~20      60      6'   y      20     4      lO'   ^z      20      15      12' 
.-.  a;=6,  2/=10,  z  =12. 

Note  4.  The  nubiber  of  equations  greater  than  that  op 
UNKNOWN  elements.  So  many  equations  as  there  are  unknown 
elements  may  be  taken  at  random,  and  solved.  If  the  roots, 
so  found  satisfj'  the  remaining  equationSj  the  system  is  possible  ; 
but,  if  not,  the  system  is  impossible. 

E.g.,  to  find  x,  y  from  the  sj'stem  of  three  two-unknown 
equations  Zx+ly  =  ll,  6x—2y=l,  8x+y  =  10: 

Take  the  first  two  equations  and  solve  ; 
then   *.*  a;  =  l,   y  =  2   in  these  two  equations, 
and    •.*  these  roots  satisfy  the  third  equation, 

.  • .  this  system  of  equations  is  possible,  and  the  roots  are  1, 2. 
But  not  possible  is  the  system  of  equations 

3x-\-7y=17,    6x  —  2y  =  l,    8x  +  y=12. 


5,  §  7.]    SI3^IPLE  EQUATIONS,  TWO  OR  MORE  UNKNOWNS.  305 

In  general,  if  there  be  m-\-n  compatible  equations,  and  only  m 
unknown  elements,  there  are  n  equations  of  condition;  and  the 
constants  must  have  such  relations  that  these  equations  of  con- 
dition are  all  satisfied. 

E.g.,    given  the  system  of  three  two-unknown  equations 
ax-\-by  =  c,   a'x  -f-  b'y  =  c',   a"x  +  b"y  =  c"  ; 

^,  cb'  —  c'b  ac'  —  a'c  r*        ^    ^  a. 

then   '.'  x  =  ^ p-,    y  =  — ; -,  [from  first  two  eq. 

ab'—a'b  ab'—a'b 

.'.  that  a"x  +  b"y  =  c'   be  a  true  equation, 

-}-  b"  •  ^'-^'^  =  c"  must  hold  true  ; 


r 


ab'—a'b  ab'—a'b 

i.e.,  ab'c"-  ab"c'+  a'b"c  -  a'bc"-{-  a"bc'-a"b'c  =  0 

is  the  required  equation  of  condition,  and  establishes 
the  necessary  relation  between  the  given  constants. 

By  this  process  all  the  unknown  elements  are  eliminated  from 
the  given  equations.  So,  in  general,  from  n  equations  n  —  1 
unknown  elements  may  be  eliminated. 

Note  5.  The  number  of  unknown  elements  greater  than 
THAT  OP  EQUATIONS.  If  there  be  m  +  n  unknown  elements  and 
only  m  equations,  all  compatible,  to  some  n  of  these  elements  arbi- 
trary values  may  be  given,  and  the  roots  of  the  m  equations  will 
contain  these  arbitrary  values,  or  some  of  them,  and  be  them- 
selves arbitrary,  and  the  equations  are  indeterminate. 

E.g.,  to  find  x,  y  from  the  single  two-unknown  simple  equation 
2a;  +  3y=12: 

Put      2/=-  -5,  -4,  -3,  -2,  -1,  0,  +1,  +2, +3,  +4,  +5,..., 
then         a;  =  ...13i-,  12,  10|,  9,  7i,  6,  4|-,     3,  H,     O,"!!,. 
or  put      a;  =  -..  "5,  "4,  "3,  "2,  "1,  0,  +1,  +2, +3,  +4,  +5,- 
then         2/  =  ...+7i,+6f ,  +6,+5J,+4|,+4,+3i,+2f ,  +2,+H,  +J,. 

So,  a  series  of  values  are  given  to  x  increasing  by  1,  and  there 
result  a  series  of  valuers  for  y  decreasing  by  f .     This  maj^  be 
illustrated  geometrically,  by  taking  x,  y  as  the  running  co- 
ordinates of  a  point  on  a  straight  line  whose  equation  is 
2a;-f  3i/=12. 

Such  series  are  called  arithmetic  progressions.  [XII.  §  1 


oOQ  EQUATIONS.  [XI.  pr. 

If  the  results  be  limited  b}^  the  condition  that  they  shall  all 
be  integers,  or  all  positive  integers,  it  may  liappen  that  there 
are  very  few  such  roots,  and  certain  modifications  may  be  made 
in  the  method  of  solution. 

E.g.,  to  find  all  possible  pairs  of  positive  integral  roots  that 
satisfy  the  single  equation   2a;-}-o?/=12; 
then         x=G—y  —  ^y. 

Put   iy  =  z; 
then         y=2z,   x=6  —  Sz. 

Put  z  =  ...,  -3,  -2,  -1,  0,  n,  +2,  +3,  ... ; 
then  y=  ...,-6, -4, -2,  0, +2, +4, +6,  ..-, 
and  a;=  ...,  15,  12,    9,6,    3,    0,-3,.-., 

wherein   6,  0  ;  3,2;  0,  4   are  the  only  pairs  of  roots  admissible. 
The  progressions  are  here  noticeable  again ;  that  for  y  in- 
creases by  2,  and  that  for  x  decreases  by  3,  and  they  both  go  on 
either  way  forever. 

So,  to  find  sets  of  positive  integral  values  for  ic,  y,  z  that 
satisfy  the  pair  of  equations 
a;  +  2?/  +  32=22,   Sx-6y  +  2z  =  -2: 
then    -.*  II3/  + 72  =  08,  [elim.  x 

...  z  =  d-2y-\-}(5-{-Sy).  [solve  for  ;3 

Put       t=l{o-^By); 
then         y  =  2t-2-{-^(t  +  l). 

Put      ti=l(t  +  l); 
then  t  =3u  —  l. 

.-.  y=2{Bu  —  l)  —  2  +  u  =7w  — 4, 

2=9-2(7w-4)+3w-l  =16-llw, 

and  a;  =  22-2(7w-4)  -B  (U-Uu)  =19u-18. 

Put      «=...,    -3,    -2,    -1,      0,    1,    2,      3,.-.;    [dif. +1 
then         x=  ..-,  -75,  -56,  "37,  "18,    1,  20,    39,  ..-,     [dif.  +19 
2/=..., -25, -18, -11,    -4,3,10,     15,...,     [dif. +7 
z  =  ...,    49,     38,    27,     16,    5,-6,-17,...,    [dif.  "11 
and  the  only  set  of  positive  integral  roots  is  1,  3,  5. 

But       of  sets,  whereof  two  are  positive  and  one  negative, 
or  vice  versa,  there  are  an  infinite  number ; 
and  of  sets  whereof  all  three  are  ne«:ative  there  are  none. 


5,  §  8.]  GRAPHIC   EEPEESENTATION.  307 

§8.     GRAPHIC    REPRESENTATION    OF    SIMPLE    EQUATIONS 
INVOLVING   TWO   UNKNOWNS. 

Every  simple  equation  involving  two  unknown  elements  may 
be  reduced  to  the  type-form,  y  =  mx  +  h.  [VII.  §  11 

jE.c/.,  the  equation  Aa;  +  B?/  =  c  becomes  y  = 3?+-, 

wherein =  ??i,    -  =  6. 

B  '    B 

Every  such  equation  may  therefore  be  represented  by  a  straight 
line  ;  and  conversely,  every  straight  line  has  its  equation. 

E.g.^  the  equation  6  a;  -}-  T^/  =  85  reduces  to  y—  —  fa;  +  12-^, 
wherein    —  |-  =  m,  12^  =  6,  of  the  type-form. 

This  equation  is  represented  by  the  line  cd  below. 

This  figure  serves  also  to  illustrate  the  solution  of  indeterminate 
equations   [§7,  nt.  5],   wherein   a;   is  a       . 
variable  and  ?/  a  function  of  x.  \ 

If  there  be  two  simple  equations  in-  ^V. 

volving  the  same  two  variables,  and  if  it  !   \\^ 

be  required  to  find  roots  that  satisfy  both  |     \   ^\^ 

of  them,  then  the  two  loci,  platted  with      - — i- — a  \d    ^^v 
reference  to  the  same  origin,  reference-  ^ 

line,  and  scale,  will  meet  in  a  point  whose  co-ordinates  are  the 
roots  sought. 

E.g.,i£  Qx-\-7y  =  12,  2x-\-Sy  =  4:  be  a  pair  of  simultaneous 
equations  whose  loci  are  cd,  ef, 
and  if  these  loci  meet  at  p  ; 

then         the  lengths  of  the  co-ordinates  ob,  bp  are  the  common 
roots  of  the  two  equations. 

But  if  the  coefl3cients  of  the  variables  in  one  equation  be  nearly 
equal  to  those  in  the  other,  then  the  loci  are  nearly  parallel,  and 
the  point  of  intersection  may  recede  to  a  great  distance  ;  if  they 
be  identical  with  those  in  the  other,  or  equimultiples  of  them, 

then  the  two  loci  are  {  ^arallef  ^^^  ^^  *^®  absolute  term  of  the 


308  EQUATIONS.  [XI. 

be 
first  -{  V         .the  like  multiple  of  that  of  the  other ;  and  there 

,  an  infinite  number  of  - 

are  <  common  roots. 

•  no 

If  one  of  a  pair  of  equations  involving  the  same  two  variables 
be  y  =  0,  the  locus  of  this  equation  is  the  line  ox,  and  the  solu- 
tion of  the  pair  of  equations  y  =  0^  y  =  mx  +  b  reduces  to  the 
solution  of  the  single  simple  equation  involving  one  unknown 
element,  mx-\-b=0,  wherein  the  locus  of  a;,  y  is  the  point  where 

the  pair  of  lines  cross,  and  whose  co-ordinates  are ,  0. 

m 

§9.     BEZOUT'S  METHOD,  UNKNOWN  MULTIPLIERS. 

Let       aiX  +  biy  +  CiZ-\-"-=hi, 
a2X  +  b2y  +  C2Z+  ...  =7i2, 

«««  +  &„2/  +  c„z  +  ...  =  K, 

be  a  system  of  n  simple  equations  involving  any  same  n  un- 
known elements. 
Multiply  the  first  equation  by  A^i,  the  second  by  A:,,  •••,  the  ?ith 
by  ^n?  wherein  ki  =  0,  and  k^, '"  k^  are  unknown  ; 

then   'r  aiX-{-biy-{-CiZ-\- '"  =7ii, 

kzCh^  -f  k^hzy  -\-k2C2Z-\ =  kz^h, 

...  ...  ...  ...^ 

Ka„x  +  k^b^y-{-k^c„z+  ...  =k^h„, 

.'.   (ai  +  ArgaaH [-k„a^)x 

+  {h  +  k2b2-\-"'+k,b:)y 

-f  (Ci+A-aC^  H f-A:„c„)z-f--- 

=  7ii  +fc2^2H l-fc„^„. 

Put  all  the  coefficients  except  that  of  x  equal  to  0, 

i.6.,put  6i+A;2&2H hA:„6,»=0,    Ci+AjgCaH f- A;„c„=0, ..-, 

and  thus  form  a  system  of  ?i  —  1  equations  involving  the  same 
n  —  1  unknown  elements,  k^-)  •••  A;„. 

Whichever  of  A^i,  A:2,  ...  k^  be  taken  as  1,  and  the  others  as  un- 
known, the  ratios  k^ikz^'-'k^  come  out  the  same ;  but  if  A^a  or 
A^  or  ...  be  oo  when  A^i  =  1,  then  k^  or  k^  or  ••.  should  be  taken  as 
1,  and  A*i,  •••  as  unknown,  whence  \  =  0. 


§9.]         BEZOUT's  METHOD,  UNKNOWN  MULTIPLIERS.  S09 

So,  by  aid  of  the  multipliers  Zj  •••  ^„_i,  reduce  this  system  to 
a  system  of  n  —  2  equations  involving  the  same  71  —  2  unknown 
elements,  say  I2'"  ?„_i,  and  so  on  ;  and  finally  to  two  equations 
involving  two  unknown  elements,  say  ^2,  rg,  and  to  one  equa- 
tion involving  one  unknown  element,  say  t. 

Solve  this  equation  for  ^,  then  solve  for  i^-,  r^,  then  for  •••, 
then  for  Zg  •••  L-i-)  then  for  ^'2  *••  K^  then  for  x,y,z,  •••. 

E.g.,  to  find  x,  y  from  the  pair  of  equations 
ax-{-by  =  c,     a'x  +  b'y  =  c' : 
then         {a  +  ka')x  +  {b-\-W)y  =  c  +  kc'. 

Put      b-{-kb'  =  0; 

. ,              7           &        J           c  +  Tec'       cb'  —  c'6 
then         k  = and    x  =  -—^ = 

b'  a-\-  ka'     ab'  —  a'b 

So,  put  (a  +  ka')  =  0  ; 

,,  7  ct        J  c  —  kc'     ac'  —  a'c 

then         k  — and   y  = .  =  — -• 

a'  ^      b-kb'     ab'-a'b 

So,  to  find  X,  y,  z  from  the  system  of  equations 

ax-\-by+cz  =  d,   a'x-\-b'y-}-c'z  =  d\   a"x+b"y+c"z  =  d": 

then         (a+k'a'-hk"a")x+  (b-^k'b'+k"b")y  +  (c-\-k'c' +k"c")z 

=  d+k'd'+k"d". 

Put      b-{-k'b'  +  k"b"  =  0,   c  +  k'c'  +  k"c"  =  0',      [k',k" unk. 

then         b  +  hc-^  {b'  +  7ic')k'  +  (6"  +  7ic")A;"  =  0. 

Put      b"  +  hc"  =  0] 

then         k   =.-^1   and^^'  =  -^  +  ^--^"^-^^" 


So,       A;" 


6c'  -  6'c 


6'c"  -  6"c 


But    ...  0.   =d±mL^l^ 

^l,'c"-b"c'       ^b'c"-b"c' 

"^  ,    6'^c-&c^^    ,,  .    6c'-6V~^' 
^  "T  rr-;^ 777-. '  ^  "t 


2,'c"-6"c'  6'c"-6"c' 

The  reader  may  reduce  this  fraction  to  a  simple  fraction,  and 
write  the  values  of  y,  z  by  symmetry. 


310  EQUATIONS.  [XI.  pr 

§  10.    SPECIAL  PROBLEMS  OF  THE  FIRST  DEGREE. 

In  a  special  problem  certain  elements  are  given  and  certain 
other  elements  have  given  relations  to  those  first  named,  and 
are  to  be  found.  These  relations  are  the  same  whether  ex- 
pressed in  ordinary  language  or  in  symbolic  language.  If  in 
symbolic  language,  their  expression  gives  an  equation  or  a 
system  of  equations ;  and  the  elements  whose  values  are  to  be 
found  are  the  unknown  elements  of  these  equations. 

The  solution  of  a  problem  embraces  three  distinct  parts : 
(1)  putting  it  into  equation  ;  (2)  solving  the  equation  or  system 
of  equations ;  (3)  discussing  the  results  under  special  conditions. 

A  problem  is  of  the  first  degree  if  its  solution  depend  on  the 
solution  of  an  equation  or  system  of  equations  of  the  first 
degree  only. 

PrOB.  6.       To   PUT   A   SPECIAL   PROBLEM   INTO  EQUATION. 

By  careful  study  of  the  enunciation  of  tJie  problem,  ascertain 
wldcii  of  the  elements  named  in  it  are  Jcnown,  and  which  are 
unknown;  represent  both  the  known  and  the  unkiioiun  elements 
by  symbols;  and  express  in  symbolic  language  all  the  relations 
that  subsist  betiveen  them. 

TJiese  symbolic  expressions  are  the  equations  sought. 

Note  1 .  It  may  be  convenient  to  express  all  the  unknown 
elements  by  aid  of  a  single  symbol. 

E.g.,  to  divide  $6341  among  a,b,  c,  so  that  b  shall  have  $420 

more  than  a,  and  c  $560  more  than  b  : 
Put  X  for  a's  share,  x  +  420  for  b's,  a;  +  420  +  560  for  c's  ; 
then         x  +  x-\- 4:20 -\-x +  4:20 -{-  560  =  6321 ,  a  single  one-un- 
known simple  equation. 
So,  to  divide  the  number  144  into  four  parts,  such  that  the 
first  part  increased  by  5,  the  second  decreased  by  5, 
the  third  multiplied  by  5,  and  the  fourth  divided  by  5, 
shall  all  equal  the  same  number : 
Put  x  for  the  number  to  which  the  several  results  are  equal ; 
then         x  —  o-\-x-\-5  +  x:  5+2^-5  =  144. 


G,  §  10.]        SPECIAL  PROBLEMS  OF  THE  FIRST  DEGREE.        311 

Note  2.     It  may  be  conveDient  to  express  different  unknown 
elements  by  different  symbols  ;  and  to  form  a  system  of  simul- 
taneous equations  involving  two  or  more  unknown  elements. 
E.g.^  a  vintner  at  one  time  sells  20  dozen  of  port  wine,  and 
30  dozen  of  sherry,  and  for  the  whole  receives  $600 ; 
and  at  another  time  he  sells  30  dozen  of  port  and  25 
dozen  of  sherry,  at  the  same  price  as  before,  and  for 
the  whole  receives  $700. 
Put  X  for  the  price  of  a  dozen  of  port,  and  y  for  that  of  a 
dozen  of  sherry ; 
then         20  a; +  30?/ =  600,    30a;  +  25?/=  700,    a   pair   of  two- 
unknown  simple  equations. 
So,  if  a  certain  rectangular  bowling-green  were  5  yards  longer 
and  4  yards  broader,  it  would  contain  113  yards  more  ; 
but  if  it  w^ere  4  yards  longer  and  5  yards  broader,  it 
would  contain  116  yards  more. 
Put  X,  y  for  the  length  and  breadth  ; 
then  (a;  +  5).(7/  +  4)  =  a;^+113,  {x+4.)'{y +b)=xy +UQ, 

So,  if  A,  B,  c,  D  engage  to  do  a  certain  piece  of  work  ;  if 
A,  B  together  can  do  it  in  12  days ;  a,  d  in  15  days ; 
c,  D  in  18  days ;  and  if  b,  c  begin  the  work,  after  3 
days  A  joins  them,  after  4  days  more  d  joins  them,  and 
all  working  together  they  finish  it  in  2  days,  in  what 
time  can  each  man  do  it  working  alone  ? 
Put  a;,  2/5  2^,  u  for  the  number  of  days  needed  by  a,  b,  c,  d  ; 

.,  1,1      1      1,1      1      1,1      1      9,9,6,2      , 

then         _+-=         _+_=         __}__=         __^__f__4-_  =  i^ 

X  y  12  X  u  \o  z  u  IS  y  z  x  u 
a  system  of  four  simple  four-unknown  equations. 
Note  3.  Discussion  of  the  Solution:  To  discuss  the  solu- 
tion of  a  problem  whose  answer  is  numerical,  is  to  try  whether 
all  the  conditions  of  the  problem  are  satisfied  by  all  or  any  of 
the  numbers  that  are  found  to  satisfy  the  equations  into  which 
the  problem  was  translated  ;  and,  if  not,  to  observe  what  other 
conditions  the  unknown  elements  must  satisfy  besides  those 
taken  account  of  in  putting  the  problem  into  equation. 


312  EQUATIONS.  [XI.  prs. 

To  discuss  the  solution  of  a  problem  whose  answer  is  literal  is 
to  observe  between  what  limiting  numerical  values  of  the  known 
elements  the  problem  is  possible ;  and  whether  any  singulari- 
ties or  remarkable  circumstances  occur  within  these  limits. 

E.g.,  in  a  certain  two-digit  number  the  first  digit  is  half  the 
other,  and  if  27  be  added  to  the  number,  the  order  of 
the  digits  is  reversed  ;  what  is  the  number  ? 

Put  X  for  first  digit,   y  for  second  digit ; 

then   •.•  2a;  =  ?/,    10a;  +  2/-f-27  =  10y+ a?, 

.-.  a;  =  3,    2/  =  6,    the  number  is  36  ;     and36  +  27  =  63. 

"Were  this  the  statement :  in  a  certain  two-digit  number,  the 
first  digit  is  half  the  other,  and  if  24  be  added  to  the  number, 
the  order  of  the  digits  is  reversed ; 

then   •••  2x  =  ?/,    10a;  +  2/+ 24  =  lOy  +  a;, 

.-.  x  =  2 J,    y  =  5 J,    and  the  number  is  impossible. 

The  statement  of  the  problem  puts  a  limitation  upon  the  values 
of  a;,  y  not  expressed  by  the  equation  :  they  must  be  integers. 

Were  this  the  statement :  of  two  numbers  the  first  is  half 
the  second,  and  if  to  ten  times  the  first  the  second  and  24  be 
added,  the  sum  is  the  sum  of  ten  times  the  second  added  to  the 
first ;  then  the  same  equations  as  before  would  express  the  rela- 
tions, and  the  values  2J,  5^  would  satisfy  all  the  conditions. 

For       2.2f  =  5i,    10.2|-t-5^  +  24  =  10.5^  + 2J. 

And  were  this  the  statement :  in  a  certain  two-digit  number 
the  first  digit  is  half  the  other,  and  if  a  be  added  to  the  num- 
ber, the  order  of  the  digits  is  reversed  ; 

then         2x  =  y^    lOx  +  y -\- a=\Qy  -\-x^    a;=i.a,    2/  =  |«; 

the  special  condition  is  imposed  that  a  shall  be  a  multiple  of  9 
not  greater  than  36  nor  less  than  —  36  ; 

i.e.,  a  is  36,    27,    18,      9,    0,      "9,    "18,    "27,    -36, 

and  the  number  is  48,    36,    24,    12,    0,    "12,    "24,    "36,    "48. 


6-8,  §11.]   QUADRATIC   EQUATIONS,    ONE   UNKNOWN.  813 

§11.     QUADRATIC   EQUATIONS   INVOLVING  ONE    UNKNOWN. 

PrOB.   7.       To    SOLVE    AN   INCOMPLETE    QUADRATIC    EQUATION. 

Reduce  the  equation  to   the  type-form  x^  =  q,   and  take  the 
square  root  of  both  members;  then   x  =  ±  ^q. 
E.g.^    to  find  x  from  the  equation 

j(a;2_io)  +  Jg {Qa?-  100)  =  3ar^-  65  : 

then   •.•  lOa^  -  100 +  18aj2_  300  =  90a^- 1950,     [mult,  by  30 
.-.  -62a2  =  -1550, 
.-.  ar^=25    and   aj=±5. 

Note.  There  are  two  square  roots,  opposites  of  each  other ; 
they  are  both  real  if  the  q  of  the  type-form  be  positive,  and 
both  imaginary  if  the  q  be  negative. 

PrOB.    8.       To    SOLVE   A   COMPLETE   QUADRATIC    EQUATION. 

Reduce  the  equation  to  the  type-form  x^  +  px  =  q. 

Add  -^p^  to  both  members  of  the  equation;  take  the  square  root; 
and  solve  the  equations  thus  found. 

The  result  is  of  the  form   x  =  —  Jp±i-y^(p^+4q). 

E.g. ,    to  find  X  from  the  equation   3a5^  +  9a;=120: 
then   •.•  ic2  +  8a;  =  40,  [div.byS 

.'.  x^+Sx -{-2^  =  421,  .  [add(f)S=2J 

.*.  a;  +  1^  =  ±  6^,  [extr.  sqr.  rts.  of  both  mem. 

.-.  a;  =  — 11  ±  6i  =  5  or  —  8  ;  and  5,  8  are  both  roots. 

So,  to  find  X  from  the  equation    ax^  -\-bx-\-c  =  0: 

then        ^  +  h  +  ^^  =  ^^zA^, 
a        4a^  4  a^ 

and  X  =  — —     V  V — H — 9^  -    and  both  values  are  roots. 

2a 

Note  1.    Double  Signs  :    Since  either  x-\-p  or  —(x-\-p)  is 

a  square  root  of  a^-|-jpic  + Jp^,  the  given  quadratic  is  satisfied 

as  well  when 

-  (ic+jp)  = -1^(2)24- 4 g)   as  when  x-{-p=i^(p^-^  4:q); 

but  this  gives  only  the  two  values  for  x  written  above. 


314  EQUATIONS.  [XI.  pr. 

Note  2.     Discussion   of   the   equation    x^-\-px  =  q^    four 
SPECIAL  CASES.     The  roots  are  : 

(a)    p  positive,  q  negative. 

Two  real  roots,  both  negative,  if  p--|-4^  be  positive. 

Two  real  roots,  both  negative,  equal  to  —  ^p,  if  j9^  +  4g  =  0. 

Two  imaginary  roots,  conjugates,  if^-4-4g  be  negative. 

(6)  p,  q  both  negative. 
.    Two  real  roots,  both  positive,  if  |)^-|-4g  be  positive. 

Two  real  roots,  both  positive,  equal  to  —  ^p,  if  p^-f-4Q'  =  0. 
Two  imaginary  roots,  conjugates,  if  p^  -f-  4  (?  be  negative. 

(c)  p,  q  both  positive. 

Two  real  roots,  the  smaller  positive,  the  larger  negative. 

(d)  p  negative,  q  positive. 

Two  real  roots,  the  smaller  negative,  the  larger  positive. 

Note  3.    Sums  and  products  of  roots.     The  sum  of  the 
two  roots  is  —  p,  and  their  product  is  —q. 
The  reader  may  prove. 

Note  4.    The  absolute  term,  0.      If  g  =  0,   then   of   the 
equation   x^  -\-px  =  0   the  two  roots  are  0  and  —  p,  both  real. 
Note  5.    Solution  by  factoring.     Write  the  equation 
x^ -^-px  —  q=0    in  the  form 
x'+px-\-ip'-i(p'-{-4q)  =  0, 
I.e.,  in  the  form    (x  +  ^pY  —  i(p-+  4  g)  =  0  ; 

then   •.•   [x-\-^p-.i^{f-^4.q)-].[x-hip-\-^^(p'+4:q)']  =  0, 
and     *.•  this  product  vanishes  when,  and  only  when,  one  of  its 
factors  vanishes, 
.-.  the  roots  of  the  equations 
^  +  ^P-iV(P'  +  4g)  =  0 
and  a;  +  ii)  +  |-V(i>'  +  4g)  =  0  [th.4cr.l 

are  the  roots  of  the  given  equation. 
...  a;=-ip  +  iV(P'  +  4g), 


8,  §11.]        QUADRATIC  EQUATIONS,  ONE  UNKNOWN.  315 

In  practice  the  factoring  is  often  made  at  sight. 
E.g. ,  to  find  x  from  the  equation  a;^  —  5a;-f-6  =  0: 
then    '.'Qi?—6x  +  Q—{x  —  2){x  —  2>),  [factoring 

.♦.  the  roots  are  2  and  3. 

Note  6.  General  rule.  The  rule  for  solving  incomplete 
quadratic  equations  may  be  stated  in  a  more  general  form  : 

Reduce  the  equation  to  the  type-form  ax^  -f  bx  4-  c  =  0. 

Multiply  {or  divide)  both  members  of  the  equation  by  a,  or  by 
any  factor  or  multiple  of  a  that  shall  make  the  coefficient  of  the 
first  term  a  perfect  square. 

Add  to  both  tnembers  of  the  equation  whatever  is  necessary  to 
make  the  first  member  a  perfect  square^  and  take  the  square  root. 

Solve  the  simple  equations  thus  found. 

The  rule  in  this  form  often  avoids  fractions.  Both  rules  rest 
on  that  for  finding  the  square  root,  and  are  the  same  in  principle. 

The  solution  of  the  equation  gives  x  =  — "^ — — • 

The  reader  may  translate  this  formula  into  a  working  rule  for 
finding  the  value  of  x  without  writing  the  intermediate  steps. 

E.g.,  to  find  x  from  the  equation   3a;^  +  9 a;  =  120  : 

[mult,  by  3 

[sq.  rt.  of  1st  mem. 


then   • 

.•  9a;2_^27a;  =  360, 

and 

9ar'-f  27a;|3a;  +  4^ 

6a; +  4^  27a; 

27a;  +  20J 

. 

..  3a;-f-4i=V380J  = 

±19i, 

and 

x=5   or    —  8  ; 

or,  by 

direct  substitution  in  the  formula, 

^_    -9±V(9^-4 

.3.-120)_ 

5  or   -8. 
2.3 


316  EQUATIONS.  [XI.  tlis. 

Note  7.     Discussion  of  the  equation    aa^-}-6a?  +  c  =  0 ; 

THREE    SPECIAL   CASES  :    C  =  0,     6  =  0,     tt  =  0. 

(a)  If  c^  the  absolute  term^  be  0  ; 
then         the  equation  ax^-\-bx  =  0  gives  x=0  and  x=—b:a, 
two  real  roots,  whereof  one  is  0. 
(6)  i/*b,  the  coefficient  of  the  first  power  ofx,  be  0  ; 
then         the  equation    aar^-|-c  =  0   gives  x=±^{—c:  a),   two 
real  roots,  opposites,  if  a,  c  be  of  contrary  signs ;  two 
imaginary  roots,  conjugates,  if  a,  c  be  of  the  same  sign. 
(c)  If  a,  the  coefficient  of  the  second  power  ofx^  be  0  ; 
then   •.•  ^^-&  +  V(;-^-4ac)^       _6_^^(6^_4ac) ^ 
2a  2a 

.-.  a;  =  (-5-hV^'):0,     (-6-VZ>'):0;  [a  =  0 

I.e.,      ■    a;  =  -0:0,    26:0      if  V^'  ^^   +  6 ; 
and  a;  =  -26:0,    0:0     if  V62  be  -  6. 

In  either  case  there  is  an  infinite  and  an  indeterminate  root. 
But  this  indeterminate  root  may  be  determined. 
For     •••  a  =  0, 

.*.  when  aj^fcoc,  the  equatiop    aa:^  +  6a;  +  c  =  0   becomes 
6x  -f  c  =  0,    whose  single  root  is  — c  :  6. 
It  may  also  be  determined  by  multiplying  both  terms  of  the 
fraction    -^^  V(^'-^^^)   by    _6  t  V(2^'-4ac)  ; 

then         X-      '  b'-(b'-4a^)  ^  2c 

2a[-6q:V(^'-4«c)]       -b  :f  ^{b' -4.ac) 
z=z  —  c'.b   or    — c:0   when   a  =  0. 
This  case  is  especially  important  as  showing  the  value  of  the 
limits  of  the  roots  of  the  equation  when  a  ==  0  ;  and  it  is  to  be 
noted  that  as  a  =  0  one  of  the  roots  =  oo,  and  the  other  =  —  c  :  6. 
This  is  also  evident  if  the  equation  be  written  in  the  form 

a;~\6  +  cx~'^)  =  —  a.       [div.  eq.  6a;  +  c  =  —  aa^  by  a;^ 
For,  if  a  =  0, 
then         either  a;~^  =  0,    and    a;  =  cx), 
or  6  +  ca;~^      =0,    and   a;  =  — c:6; 

i.e.,  both  00   and  —  c  :  6  satisfy  the  equation  and  are  roots. 


8,  §11.]         QUADRATIC  EQUATIONS,  ONE  UNKNOWN.  317 

To  determine  whether  the  root    =  +oo  or  ~oo,  observe  that 

•••  the  sum  aocr-^bx,  =— c,  remains  finite  when  ax^  and 

bx  each  =  co,    i.e.,  when  ic=  oo, 

.*.  aa^,  bx  have  opposite  signs. 

Divide  aa;^,  bx  by  ax ; 

then   *.*  X  and  b:a  have  opposite  signs, 

•  «        y^  X,         I  the  same     .  ,,  .     .    i  ~oo 

.-.  If  a,  6  have  ^  ^p^^^j^^   signs,  the  root  =  ^  +^ 

when  a  =  0. 
The  reader  may  further  discuss  the  equation  aoiy^-\-bx.-\-c  =  0, 
after  the  manner  of  Note  2,  and  show  that  the  two  roots  are 
real  and  unequal  I  62>4ac.' 

real  and  equal      when   -{6^  =  4ac. 
imaginary  |  6^  <  4  ac. 

Of  the  real  and  unequal  roots  he  may  show  which  is  the 
larger ;  and  of  all  real  roots  he  may  show  the  conditions  that 
make  them  positive  or  negative.  He  may  also  show  that  in 
every  case  the  sum  of  the  two  roots  is  —b:a,  and  their  product 
c :  a,  and  that  if  x',  x"  stand  for  the  two  roots, 
then         ax^  -\-  bx  -^  c  =  a  {x  —  x')  (x  —  x") . 

Note  8.  Equations  solved  as  quadratics  :  Every  equation 
of  either  of  the  following  forms,  or  reducible  thereto,  is  solved 
by  aid  of  quadratics  : 

(a)       ax2«4-5a?"-|-c  =  0, 

(6)        (aa^™  +  bx''  +  c)^'"  -|-p(aaj2n  _j_  j^n  _|_  (.)»«-}-  g  =  0, 

(c)        (aa^«  +  &a;«-f  c)2'»±(ea;'»+/)2'"  =  0, 

{d)       (aa^'*  +  6a;'»  +  c)2"'±((ii»2»»4-ea;'^)2"'=:0, 
wherein  a,  6,  c,  d,  e,/,  p,  q  are  independent  of  ic,  and  may  be 
real  or  imaginar}'. 

Whether  a  given  equation  p  =  0,  whose  degree  is  eve'h,  be  of 
form  (a),  appears  at  once.  If  it  be  not,  then  to  see  whether  it 
reduces  either  to  form  {b)  or  to  form  (c) ,  find  r,  the  entire  part 
of  the  square  root  of  p  :  if  the  remainder  p  —  r^  be  of  the  form 
pR  +  g,  the  equation  reduces  to  R  =  i[— i:>+ V(P^~  ^^)]'  ^^^ 
if  also  R  or  some  root  of  r  bo  of  the  form  a^""  -^bx""  -\-c,  the 
equation  reduces  further  to  (b) ;    or  if  p  —  r^  be  ±  a  perfect 


818 


EQUATIONS. 


[XI. 


square,  S",  the  equation  reduces  to  r  =  s ^'''1?  ^^^  perhaps  to  (c). 
Or,  arrange  p  to  ascending  powers  of  x,  and  find  r',  so  much 
of  -y/F  that  r'^  has  the  degree  of  p  as  to  a; ;  then  if  p  ^  r'^  be 
±  a  perfect  square,  s'^,  the  given  equation  reduces  to  R=s'-y/'^l, 
and  perhaps  to  (d) . 

E.g.,  irdx*-52a^  +  64:  =  0; 
then   •••  81a;*-468a^4-C76  =  100,  [mult,  by  9,  add  100 

.-.  9ic2^26±10, 
.♦.  ar^    =4        or  -y, 
.'.  X     =  ±  2   or    ±^:  four  real  roots. 
So,  if  (9a;^-o2a:2_|_go)24.9(9a;4_52a;2_^g0)  _  400  =  0  ; 

then   •.•  4(9a;*-52ar4-80)2+3G(9a;4-52a^4-80)+81  =  1681, 
.-.  2(9a;^-52ar+80)  =  -9±41 

=  32  or  -50, 
.-.  9a;*-52ar  +  80  =16  or  -25, 
.'.  x=±2,    ±|,    ±  iV(26±V- 209)  ;  eight  roots. 


§12.    GRAPHIC   REPRESENTATION   OF   QUADRATIC 
FUNCTIONS. 

Let  ax^  +  bx-{-c  be  an^'  quadratic  function  of  a?,  and  put  y 
equal  to  it ;  then  different  values  may  be  given  to  a?,  the  cor- 
responding values  of  y  computed,  and  the  function  platted.  The 
plat  is  a  parabola  whose  axis  is  vertical. 


E.g.^  in  the  equation   y=za^-\-2x  —  3. 

Put  a;=...,  -5, -4, -3, -2,  -1,  0,  +1,  +2,  +3,  +4,  +5,...; 
then  y="',12,  5,  0,-3,-4,-3,  0,  5,12,21,32,..., 
and  the  plat  of  the  function  is  as  shown  in  the  figure,  p.  319. 

If  there  be  a  pair  of  equations  involving  x,y  y  =  x^  -{-2x—3, 
y  =  0,  their  solution  is  reduced  to  the  solution  of  a  single  quad- 


§12.] 


QUADRATIC   FUNCTIONS. 


319 


ratic  equation  involving  one  unknown  element,  x^-^2x—S  =  0 

and  the  roots  of  this  equation  are  the 

abscissas  of  the  points  where  the  curve 

whose  equation  is  y=x^-\~2x—S  cuts 

the  axis  of  abscissas  whose  equation 

is    y  =  0. 

The  ordinates  of  the  points  of  inter- 
section are  manifestly  0. 

If  the  curve  that  represents  the  equa- 
tion y  =  x^-\-2x  —  S  remain  fixed  on 
the  paper  while  the  horizontal  line  that 
represents  the  equation  y  =  0  moves 
downwards,  taking  in   succession  the 

positions  o'x',  o"x",  •••,  each  ordinate  of  the  curve  is  increased 
by  the  same  length,  and  the  value  of  y  in  the  given  equation 
is  increased  by  the  same  number  ;  and,  by  the  simple  change  of 
the  absolute  term,  the  two  roots  of  a  quadratic  equation  may  ap- 
proach each  other,  then  become  equal,  then  imaginary. 

E.g.,    of  a^-^2x=S  the  two  roots  are  —3,      1, 

of  x^-\-  2a;  =  0  the  two  roots  are  —2,      0, 

of  fl^-f-2a;  =  — 1  the  two  roots  are  —1,  —1, 

of  a^+2a;  =  — 2  the  two  roots  are 

-1  +  V-i,  -1-V-i- 

In  all  such  cases  it  is  said  that  a  straight  line  cuts  the 
curve  in  two  points,  real  and  separate,  real  and  coincident, 
or  imaginary,  just  as  it  is  said  that  every  quadratic  equation 
has  two  roots,  reUl  and  unequal,  real  and  equal,  or  imaginary- ; 
and  though  it  may  seem  strange  to  the  beginner  to  say  that  one 
line  cuts  another  in  two  points  when  it  only  touches  it,  or  to  say 
that  it  cuts  it  in  two  points  when  it  does  not  cut  it  at  all,  yet 
the  language  and  the  demonstrations  of  Algebra  gain  greatly 
by  this  generality  ;  and  the  pairs  of  roots  so  described  have  most 
of  the  algebraic  properties  of  other  pairs  of  roots  :  in  particular, 
they  each  satisfy  the  given  equation,  and  their  sum  is  the  — j9 
and  their  product  the  —  g  of  the  type-form.  [ 


320  EQUATIONS.  [XI.  pr. 

PrOB.  9.      To   PLAT   THE    EQUATION     QX^  +  bxy -\- Cy^=:  d,    USING 
NO    IIIRATIONAL    FUNCTIONS   OF   X^  y  :       a,  6,  C,  d,  iC,  y,  ALL   llEAL. 

(a)  When6^>4oc,    and  cd{^  0. 


Compute  M,  =  ^  — — - —  ;    and  n,  =  ^  - 
\b2— 4ac  \ 


c 

To  V,  aw  auxiliary  variable^  give  any  convenient  series  of  val- 
ues; and  for  each  value  o/v  find  a  pair  of  simultaneous  values 
ofx,  y  to  satisfy  the  given  equation : 

2v  *^         2v  2v       c 

Plat  each  of  the  points  x,  y  ;  and  join  them  by  a  curve. 

(6)  When  4ac>b^:   then  always  cd>0.  [cc,  ?/ real 

Compute  m',  =  J*" — ;    and  n',  =  J"— 

\4ac  — b^  \c 

To  the  auxiliary  variable  v  give  any  convenient  series  of  values; 

and  for  each  of  them  find  values  of  x,j  to  satisfy  the  equation: 

2v      ,-,    ,            1-v^     ,        2v      bM' 
VIZ..        x  = -^'2:m'.    y  = --n' 

l+v^         '    -^      l+v2  1+v'     c 

Plat  each  of  the  points  x,  y  ;   and  join  them  by  a  curve. 

(c)   When  6^  =  4ac:   then  always  c(Z<0.  [ic,  2/ real 

Compute  n',  =  ^1-  ;    i/ie  pZa^  is  iwo  parallel  straight  lines 

y  =  N' X,     y  =  — n' X. 

^  2c         -^  2c 

(a)'.'  {2cy-{-bxy—(b^  —  4:ac)'X^  =  4:cd,  [glv.  eq. 

.-.   l2cy-\-bx-{-x^(b^-4ac)}[2cy-\-bx-x^{b^-4:ac)'] 

=  4cd ; 

.*.  whatever  value  be  given  to  v, 

when        2cy -{-bx  +  x^(b"—4:ac)=     2^{±cd)'V, 

then         2cy-{-bx  —  x^{b-—4:ac)  =  ±2^{±cd):v, 

.'.  x  =  — — — ^  — =  — =i— -  .m;  Q.E.D.[elim.?/, solve 

V        \¥  —  4:ac         V 

V"  i  1 
and     •••  4:cy-^2bx  =  2 ^{±cd),  [add  eqs.  above 


9,  §  12.]  QUADEATIC   FUNCTIONS.  S21 

V^  ±1  t'^  IP  1    5  M  r       1  1   * 

.'.  y  = N — —        Q.E.D.  |_repi.  it',  sol.  for  2/ 

2 1'  2v       c 

(b)  '.'   {2cy -\-bxy=4:cd  —  {4:ac  —  b-)'X^  [giv.  eq. 

=  [2 Vc(^  +  a;  V(4ac  -  6^)]  [2  y'cc^  -  a;y'(4aG-62)]; 

.'.  whatever  value  be  given  to  v, 

when        2  ^cd  +  x-W{4:ac-b^)  =  i^  (2cy  +  bx), 

1  —  v 

then         2^cd-x^{4ac-b')  =  ^^^  {2cy  +  bx), 

...  2Vcd+a.V(4ac-6^)  ^  /l  +  ^V,  [divide 

...  x^-^^J      '^     ,     =^.2m';  Q.E.D.  [soLfora: 
and     '. •  4  y'cd  =  — ^^^- — -  (2 cy  +  6a;) ,  [add  eqs.  above 


1—v 


.-.  2/=^^=^'-n'-^^,-— •     Q.E.D.     [repl.a;,sol.for2/ 
1  +  ^'2  1  +  2;^     c 

And    •.•  4cd  is  the  sum  of  the  positive  quantities 
{2cy-^bx)\     {4:ac-b^)'X', 

.-.    Cd<0.  Q.E.D. 

(c)   •.•  when  (2c2/  +  6a;)^  =  4c(i 

then  (2c2/  +  6a;  —  2  -^cd)  {2cy+  bx  +  2  -^cd)  =  0, 

.,,  Id      6a;  Id      6a; 

...either  2/  =  ^---    or    y^-^--^^, 

and  conversely.  q.e.d. 

E.g.,  to  plat  the  equation   3a^  +  5a;?/  + 7/=  425.  [fig.,p.322 
Here    a,  6,  c,  d  =  3,  5,  7,  425  ;    4ac-6-=+59, 

and  the  case  is  (6)  ; 
m'=  V(7  •  425  :  59)  =  7.101,    n'=  V(^25  :  7)  =  7.792, 

6  m' 


=  5.072; 


and  the  coefficients  of  2  m',  n',  —  ^in  the  values  of  x,  y  are  : 


322                                             EQUATIONS.  [Xl.prs. 

When  V        =0  li  ^  ^    ^  =^1    ^  ^2   ^3   ^5    00   ..., 

5     3     2      3  2 

then        -1:^  =  0  !5  15  ^  ^2  ., 

l+ir^          13    5     5     13  13  5     5    13 

,           1--?/^      ,    12    4     3      5  ^     -5  -3   -4-12_, 

and          -  =  1   —    -     -     —  0     —    — 1   ..., 

1+172          13    5     5     13  13  5     5     13 

and  2/  =  (l,j|,-,^,-l)x7.792-(0,^,...,^,0)x5.072, 

I.e.,          a=     0         =^5.46     -8.52  ^11.36  *13.11  ^14.20..., 

.r.A          ..       - -Q     I  0.24    .  3.19  .0.62  .-1.68     t.  n- 

and          y=    /./9    \  g  j^  \  9  28  ^  8.73  <    7.68       ^'^^ -' 

=^1          -1  =^1  H 

for            v=     0            J:           _i  J.  _f            ±1     ..., 

5            3  2  3 


,,_---o     ,-9.14    .-9.28     .-8.73     ."7.68 
y-    t'^-f    i-5.24  1-3.19     '»-0.63    ^    1.68 


i.O 


and 

for  V  =     00         =^5  *3 

2 

Give  the  same  coefficients  (except  in  order  and  sign)  an}-  values 

Vi,V2  of  27  such  that  ^2=  —v^,  or  =  ±  1 :  ^i, or  =  ±  {l—v-^  :  (1  +"^1), 

or   =  ±  (1  +  Vi)  ;  (1  —  Vj) .      Such  values  are  0,  00 ,  ^1  ; 

and   ±i,    ±|,    ±2,    ±3;    and  ±|,   ±  |    ±  |,    ±5. 

So,  to  plat  the  equation 

3 o^  4. 5 a;?/  4-  7 2/2  -  14 a;  -  51 2/  =  330  : 
here         3  (a;  +  1)^  +  5(a;  +  1)  (2/  -  4)  +  7(2/  -  4)^  =  425, 
and  the  plat  is  as  in  the  above  example,  except  that  the  origin 
or  datum-point  to  which  the  curve  is  referred  will  now  be  a  unit 
to  the  right  of,  and  4  units  below,  the  origin  of  the  former  plat. 

Note.  When  cZ  =  0,  no  auxiliary  v  is  needed :  plat  (a)  re- 
duces to  a  pair  of  lines  y —  \\_—h  + ^{lr—A:ac)~\c~'^x  and 
2/  =  ^[— 6  —  V(6"— 4«c)]c-^'«;  plat  (6)  to  a  point  a;=0,  y=0\ 
plat  (c)  to  two  coincident  lines    y  =  —  \hc~^x. 


9, 10,  §13.]     SOLUTION  BY  CONTINUED   FRACTIONS.  323 

§13.   SOLUTION  OF  QUADRATIC   EQUATIONS  BY  AID  OF 
CONTINUED   FRACTIONS. 

PrOB.  10.      To  SOLVE  A  QUADRATIC    EQUATION  BY  AID   OF   CON- 
TINUED  FRACTIONS. 

First  root :  Write  the  equation  in  the  form  x{p-\-x)  =  q\ 

and  the  convergents  are  :  ^~ 

q        pq         p-q  +  q^         fq-\-2pq^     _ 


p     p^  +  q     p^  +  2jpg     p^+^p-q  +  q^ 

Second  root:  Write  the  equation  in  the  form  x^^—px+q. 

q  q  q 

then         a'  =  -i>4--  =  -p-^^2  =  -p-^p7  ^g=-, 

and  the  convergents  are  : 

_        _2^_±q         p^-\-2pq         p^-^3p'q-\-q^  ^ 
^'  p     '  p'  +  q'  p'  +  2pq 

Of  these  two  sets  of  convergents,  when  taken  two  and  two  in 
order,  the  products  are  —  g,  and  the  sums  approach  —p. 
E.g.,  to  find  x  from  the  equation  ic^  +  5ic  =  2  ; 


0                                           2 
then  the  two  roots  are   —  2               and  —  5  —  -—  2 

and  the  convergents  are  :            o -j 

*•*> 

2     10      54      290               ,      .         27         145 
6'    27'    145'    779'  •*•' '''''^      ^'         5  '    ~  27  ' 

779 
145' 

whose  products  taken  two  and  two  are  all  —2, 

and  whose  sums  so  taken  are   —  4f,  —  ^y^j^  -^^llih  " 

"j 

that  differ  from  -5  by   ?,   -|^,    ^,  •••• 

0       loO       OviO 

The  reader  may  find  the  approximate  values  of  x  from  the 
equation  aoc^ -\- bx -i- c  =  0 ,  and  translate  the  formulae  so  found 
into  words.  In  particular,  he  may  find  the  approximate  values 
of  X  when  a  =  0  ;  and  show  how  the  convergence  of  the  con- 
tinued fraction  depends  upon  the  reality  of  the  roots.  This  form 
of  solution  by  continued  fractions  applies  only  to  quadratics ; 
another  form  is  given  in  XIII. 


824  EQUATIONS.  [XI.  th.  8, 

§  14.    -MAXIMA  AND  MINIMA. 

If  a;  be  a  variable,  and  y  he  a.  function  of  x,  and  if  as  x  in- 
creases, y  increase  for  a  time  and  then  decrease,  the  greatest 
value  that  y  thus  attains  is  a  maximum  ;  but  if  as  x  increases,  y 
decrease  for  a  time  and  then  increase,  the  least  value  that  y  thus 
attains  is  a  minimum.     So  for  any  two  variables. 

The  normal  {  ™t^^^°^^  of  any  function  u  of  one  or  more  in- 
'  mmima  -^ 

dependent  variables  x^y, ..,  are  such  values  of  u  that,  if  u  were 
a  little  -j  p^^^^^'  some  of  x,  y,  ...  would  become  imaginary: 
they  depend  upon  a;,  y,  ...  being  restricted  to  real  values.  Ab- 
normal  maxima  and  minima  arise  from  other  restrictions :  as  in 
the  example  below,  where  a  certain  rectangle  is  restricted  to 
have  its  corners  at  or  between  the  vertices  of  a  certain  triangle. 
If,  by  solving  a  quadratic  or  otherwise,  the  relation  of  u  to 
a,  y, ...  be  expressed  in  the  form  p  +  Q  V^  =  ^'  wherein  p,  Q  are 
rational  functions  of  a;,  ?/,  ...  and  u  a  function  of  u ;  then  is  w  a 

normal  i  ^^^}^^^  whenever  its  value  is  such  that  u  vanishes 
^  i^]f^^f^        .  .   .  ,    ,  .    ,  a  decreasing 

and  is  not  itself  a  maxunum  or  minimum,  but  is  •{  ^^  increasing 

function  of  w  ;  for  a  slight  f m'ther  \  ^a^I^^^'^q  '^^  the  value  of  u 
makes  -^u  imaginary,  while  its  equal  — p  :  Q,  a  rational  function 
of  x^y^  . . . ,  remains  real. 

Theor.  8.   Maximum  and  minimum  values  of  a  continuous 
function  occur  alternately. 
For    *.*  just  after  passing  through  the  first  maximum  value  the 

function  is  decreasing, 
and  just  before  passing  through  the  second  maximum  value 

the  function  is  increasing  ;  [df.  max.,  min. 

and     *.*  in  passing  from  a  decreasing  state  to  an  increasing  state 
the  function  must  pass  through  a  minimum  value  ;  [df . 
.*.  between  two  maximum  values  lies  at  least  one  minimum 
value.  Q.E.D. 

So        between  two  minimum  values  lies  at  least  one  maximum 
value.  Q.E.D. 


pr.  11,  §  14.] 


MAXIMA   AND   MINIMA. 


325 


PrOB.   11.      To    DETERMINE    MAXIMA    AND    MINIMA    BY    SOLVING 
QUADRATIC   EQUATIONS. 

By  an  equation  express  the  relation  between  x,  a  variable^  and 
u,  a  function  of  x  to  he  maximized  or  minimized. 

Solve  for  x  ;  and  if  the  value  ofx  thus  found  involve  an  even 
root  of  a  function  o/u,  equate  that  function  to  0  and  solve  for  u. 
See  whether  the  values  of  u  so  found  he  maxima  or  minima. 
E.g.,  in  a  triangle  to  inscribe  a  rectangle  of  maximum  area  : 
1.   There  is  such  a  rectangle. 
For  let  ABC  be  any  triangle,  an  its 
altitude,  bc  its  base,  defg 
any  rectangle  inscribed  in  it. 
then   •.•  DEFG  approaches  a  minimum 
value,  zero,  as  gf  approaches 


value,  zero,  as  gf  approaches  bc, 
for  some  intermediate  position  of  gf  there  is  a  maxi- 
mum value  of  DEFG.  [th.  8 
2.    Let  w  =  area  DEFG,    aj  =  DE,    y  =  T>G,    &  =  bc,    ^  =  an; 
then   '.'  xy  =  u,            x:h  —  y  =  b:  h, 

...  u  =hy{h-y):h,     y  =  ^h  ±^^[{bh' ^4:hu):h], 
.*.  the  maximum  value  of  u  is  \hh,  and  gf  lies  half-way 
between  the  vertex  and  base. 
So,  about  a  sphere  to  circumscribe  a  cone  of  minimum  volume  : 
1.  There  is  such  a  cone. 
For  let  DEF  be  any  circle  and  abc 
an  isosceles  triangle  cir- 
cumscribed about  it  and 
tangent  to  it  at  d,  e,  f  ;  let 
ad  be  the  axis  of  the  tri- 
angle, and  let  the  whole 
figure  revolve  about  ad  ; 
then   •.*  as    the    point  a   recedes 
from  the  circle,  the  lines 
ab,  AC    approach    parallelism,   and    the    triangle   abc 
grows  larger  and  larger  without  bounds, 
.*.  the  cone  abc  grows-larger  and  larger  without  bounds. 


326 


EQUATIONS. 


[XL  pr. 


And   •.*  as  the  point  a  approaches  the  ch'cle  the  lines  ab,  ac 
approach  parallelism  with  bc,  and  the  triangle  abc 
again  grows  larger  and  larger  without  bounds, 
.'.  the  cone  abc  grows  larger  and  larger  without  bounds  ; 
.-.  for  some  intermediate  position  of  a  there  is  a  minimum 
value  of  the  volume  of  the  cone.  [th.  8 

2.  Let  V  =  volume  of  cone  abc,    y  =  ad,  its  altitude, 
X  =  BD,  radius  of  base,       r  =  radius  of  sphere  ; 
then         v  =^7ra^y; 
and    •.•  ab-af  =  ad-ao,   ab^  =  ad- +  bd-,   af^  =  ao^  — of', 


and 

and 
i.e.. 


AG  =  AD  —  CD, 


[geom. 


{f-\-^)-{y-r—r^)  =  y-'{y-r)  \ 

a^y=zj^f:  {y-2r), 

v    ^i^r^y':(y-2r), 

y    =[3v±V(9^'-247rr3v)]  :27rr2; 

that  2/ be  real,    9v-<247r7^v, 

the  minimum  value  of  v  is  f  ttt^, 

the  corresponding  values  of  y,  x  are  4r,  r-^2^ 

the  minimum  circumscribed  cone  has  its  altitude  double 
the  diameter  of  the  sphere,  the  area  of  its  base  two 
great  circles,  its  volume  double  the  volume  of  the 
sphere,  and  its  whole  surface  double  that  of  the 
sphere :  as  the  reader  may  prove. 

2a;  +  21 


So,  to  ascertain  if  the  fraction 


x" 


have  any  limi- 


Let 
then   • 

i.e., 


6a;-14 
tations  on  its  value,  for  real  values  of  x : 

w  = — — ; 

x=l  +  2.y±  3V[(2/-  2)(2/  + V-)].  [sol.  for  x 

that  X  be  real,  the   factors  y  —  2  and  y  +  V"   i^ust 
have  the  same  sign, 

y  may  not  lie  between  2  and  —  ^,  but  may  have  any 

other  value, 
2  is  a  minimum  and  —  ^  a  maximum  value  of  x. 


11,  §  14.] 


MAXIMA  AND  MINIMA. 


327 


The  reader  may  plat  this  function   [a  hyper'bola],   and  the 
meaning  of  these  statements  will  then  be  clearer  to  him.* 

So,  to  find  tlie  limitations  on  the  value  of  the  quadratic  func- 
tion  aa?  -\-'bx-{-c   for  real  values  of  x : 
Let      y  =  ao?  -\-hx-\-c\ 
then   •.•  x  =  \_-h±^{W-4.ac  +  4.ay)'\'.2a, 

.'.  for  real  values  of  «,  6^— 4 ac4- 4 a?/ cannot  be  negative, 


I.e., 


,  6^— 4ac  ,1      ,  negative    ,  .    ,  positive, 

y  -\ cannot  be  <      ^...       when  a  is  ^  t'^^^y^i^y^, 

4  a  positive  >  negative, 


>      4a  4a 


.    ..     .  minimum      , 
is  its  <         .         value. 
'  maximum 


The  plat  of  the  function  is  a  parabola  whose  axis  is  vertical 

aud  vertex^  t^l^d""^  ^l"*"  «  is  "I  ^egaUVe  '  '"''^  *'''"  parabola 

<  til """'  <=»*  tl^«  a^is  of  X  when  b'-4ac  is^  ""S^}:'!^- 
'  does  '  positive. 


The  four  cases  are  represented  by  the  four  cuts  below, 
o 


c      'f          ax^  +  bx-\-c 
bo,  if  v  = ' ' —  ; 


then 


-(b-b'y) 
2{a-a'y) 


■t-  -y/l{b'^-4:a'c')y^-2  (bb'-2ac'-2a'c)y+b''-'4:ac'] 
2{a-a'y) 
.*.  that  X  be  real, 

(6'2 _  4ct'c')/  -  2(bb'  -  2  ac' -  2 a'c)y  +  6'  -  4ac  <  0. 
Write  this  quadratic  function  of  y  in  the  form 
(6'2-4a'c')(2/-a)(2/-/?), 
wherein   a,  (3  are  the  roots  of  the  equation  got  by  putting  this 

function  equal  to  0  ; 
then         three  special  cases  are  to  be  noted : 


328  EQUATIONS.  [XI.  pr. 

(a)  b '  ^  —  4  a 'c '  positive . 

If  a,  /S  be  real  and  a<)8,  then,  that  x  may  be  real,  y  must  not 

lie  between  a,  /5 ;   i.e.,  y  has  a  for  a  maximum  and  ^ 

for  a  minimum  value. 
If  a,  fi  be  real  and  equal,  or  imaginary,  the  product 

(y  —  (^){y  —  a)    is  always  positive,   and  there  is  no 

limitation  on  the  value  of  y. 
(6)  b'*  —  4  a'c'  negative. 
If  a,  ft  be  real  and  a  <  /?,  then,  that  x  may  be  real,  y  must  lie 

only  between  a,  ft ;  i.e.,  y  has  a  for  a  minimum  and  ft 

for  a  maximum  value. 
If  a,  )8  be  real  and  equal,  or  imaginary,  then  the  product 

(b'^  —  4a'c'){y —  a){y  —  ft)  is  negative,  and  no  real 

value  of  X  is  possible  except  for  the  particular  values 

y=a=ft. 
(c)  b'*  — 4a'c'  zero. 
Then  the  quadratic  function  in  y  reduces  to  the  form  py-{-q; 

and  that  x  be  real  this  function  may  not  be  negative ; 

and  if  p  be  ^  P*'"'*!:"^  'y  +  ^-  cannot  be  ^  "-^S^"^'^  ' 

^        '  negative,  ^  ^  p  '  positive ; 

I  <     Q        A  -4.    I  maximum      ,      .         Q 

•  *•  V-iZ^ ,  audits^      .   .  value  IS 

•^  '  >     p'  '  minimum  p 

Note.  It  is  sometimes  better  not  to  solve  for  the  independent 
variable,  but  to  express  in  terms  of  it  the  function  to  be  maximized 
or  minimized:  noting  that  if  a  be  a  positive  constant,  m,  n  odd  posi- 
tive integers,  <f>  an  increasing  function,  and  \}/  a  decreasing  func- 

..        .,  V  .        I  maximum,  ,  -     ,/   \ 

tion,  then  when  ?<  IS  a -(      .   .  '  so  are   u±a.  au,  w»,  cf)(u)', 

'  '  minimum,  ■>   t\  /^ 

,    ,    ,  --      ,  /   \         J  minima, 

but  ±a  —  u,  a:u,  u  »*,  \^(u)  ares 

'  '         '  rv   /         1  maxima. 

E.g.,  to  divide  a  real  number  2a  into  two  real  parts  whose 

product  is  a  maximum : 

Let  a  —  2;   and   a  +  2;  be  the  two  parts ; 

then  •.*   {a  —  z)'{a-\-z)  =  a^  —  z^, 

and    •.•  the  product  a^  —  z^  is  greatest  when  2  =  0, 

.-.  the  parts  are  a  and  a.  q.e.d. 


11,  §  14.]  MAXIMA   AND   MINIMA.  329 

There  is  no  minimum  ; 
for,  as  z  grows  larger,    o?  —  z^  grows  less  without  bounds. 

Let   a  —  z  =  x\ 
then  a-\-z  =  2a  —  x^    and  the  product   a?—z^  =  x{2a  —  x). 

To  plat  the  locus  of  the  equation    ?/-  =  cc  {2a— x)  : 

Take  ox  =  2  a,   and  on  ox  as  a  

diameter  describe  a  semi-  ^^ 

circle ;  take  b  any  point  on        / 
ox,  and  draw  bp  perpen-       / 
dicular  to  ox  ;  / 

then   *.•  Bp2=OB-Bx,  [geom.     o  a     c      B  x 

and  ?/  =  BP,    a;  =  OB,    2  a  — a;=Bx, 

.-.  tlie  semicircle  is  the  locus  sought, 
and  'if  is  greatest  when  b  is  at  the  centre, 

?'.e.,  when  ob,  bx,  bp,  each  =  a. 

So,  to  resolve  a  real  number  o?  into  two  real  positive  factors 

whose  sum  is  a  minimum  : 
Let  ic,  y  be  the  two  parts  ;  ^ 

then    •••    {x-\-yy  =  {x  —  yY  +  A:xy  =  (x  —  yy-\-^a'^^ 

.'.   (a; +  2/)  is  least  when  0?  — 2/ =  0,   i.e.,  when  x  =  y  —  a. 
There  is  no  maximum  ; 
for,  as  ic~?/  grows  larger  without  bounds,  so  does  x-{-y. 

From  these  two  examples  it  appears  that  of  all  rectangles 
with  the  same  perimeter  the  square  has  the  greatest  area,  and 
that  of  all  rectangles  with  the  same  area  the  square  has  the  least 
perimeter.  So,  often  the  same  conditions  that  make  a  variable  u 
a  maximum  or  minimum  when  some  other  variable  v  is  constant, 
also  make  v  a  maximum  or  a  minimum  when  u  is  constant. 
Other  maxima  and  minima  may  be  found  by  aid  of  the  above. 
E.g.^  to  make  — -^ —  a  maximum  :  [a,  h  positive 

Make  the  reciprocal,    ax-\-h:x,    a  minimum  ; 
then    •.•  the  product  of  ax  and  6  :  a?  is  the  constant  a6, 

.*.  their  sum  is  a  minimum  at  2^ab  ; 
I.e.,  the  given  function  is  a  maximum  at  Ja~^6~^. 


330  EQUATIONS.  [XI.  prs. 

§  15.    SIMULTANEOUS   EQUATIONS. 

PrOB.  12.  To  SOLVE  TWO  EQUATIONS  INVOLVING  THE  SAME  TWO 
UNKNOWN  ELEMENTS  WHEN  ONE  OF  THE  EQUATIONS  IS  SIMPLE. 

Eliminate  one  of  tlie  unknown  elements.  [pr.  2 

Solve  the  resultant  for  the  other  unknown  element  and  replace 
this  element  by  its  value  in  the  simple  equation. 
Solve  this  equation  for  the  first  unknown  element. 
E.g.,  to  find  the  values  of  a;,  y  from  the  pair  of  equations 
Sx  +  2y=20,    -3  a^ -\- 5xy  +  7y^  =  425  : 
then    •••  x  =  i{'20  —  2y),  [sol.  first  eq.  for  a; 

.-.  i(20-22/)2+|2/(20-27/)  +  7/=425,[repl.a;inscc.eq. 
.*.  157f-\-20y  =875, and  t/=  7  or  —8  J,  [sol.  quad,  for?/ 
.*.     3a;  +2-7    =20,    and  a;=2,  [repl.  ?/ in  first  eq. 

or  3a;-2.8J=20,    and   a;=12f, 

and  the  two  pairs  of  roots  are  :  2,  7  ;  12f,  —8^. 

That  both  pairs  of  roots  satisf}^  the  two  equations  appears  by 
direct  substitution,  and  that  there  ought  to  be  two  pairs  of  roots 
is  evident  from  the  plat. 
\  Let       «=      0,      2,      4,       C,      8,     10; 

^v,.^  then         in  the  equation   3x-\-2y  =  20 

]^>V  y=    10,-      7,      4,       1,    -2,     -5, 

.  1  Ivi  ibi^ik     ^^^  in  the  eq'n   ^of -\-bxy -\-ly^  —  ^26 

'   !   i\l   l\  2/=+7.8,   +7,+6.2,+4.9,+3.5,+1.5, 

I   1    \\\J      or  2/ =  "7.8,-8.2,-8.9, -9.2, -9.3, -8.6, 

^~~"^ — ' — ' —  \        NoteI.  The  two  pairs  of  roots  ma}' coincide. 

E.g.,  to  find  x,  y  from  the  pair  of  equations 

7a;24.6a;?/4-8/+12a;+16?/-88  =  0,  23a;+222/  =  68  : 
then  the  resultant  is   /— 22/+l  =  0,  and  cc,  ?/=2,l;  2,1. 

The  geometric  interpretation  of  the  equality  of  the  roots  is 
that  the  loci  represented  intersect  in  coincident  points;  i.e., 
that  they  are  tangent.  A  slight  change  in  either  equation  so 
changes  the  locus  that  the  points  separate  or  disappear.  Then 
the  two  roots  are  real  and  separate,  or  imaginary. 


12, 13,  §15.]  SIMULTANEOUS   EQUATIONS.  331 

Note  2.    Special  expedients  may  be  useful. 
E.g.^  to  find  x^  y  from  the  pair  of  equations 
x  +  y=l,     xy=12: 
then    •.•    (x  —  yy=(x-\-yy  —  4:xy=:l, 

,'.  x-y=±  1  ; 
and     •.•  2x={x  +  y)-\-{x-y),     2y  =  (x  +  y)  -  {x^y), 

.♦.  a;  =  4  or  3,     2/=  3  or  4; 
and  the  two  pairs  of  roots  are  :    4,3;    3,4. 

So,  to  find  oj,  y  from  the  pair  of  equations 
a^-^y^=125j     x-y==6: 
then    •.•    {x^7jy-{-{x-yy==2{x'-\-y'), 
.'.   (x-{-yy  =  225   and   x-{-y=±16, 
.'.  a;=10  or— 5,    y=6  or  —  10, 
and  the  two  pairs  of  roots  are  :    10,  5  ;   —  5,  —  10. 

PrOB.  13.  To  SOLVE  TWO  EQUATIONS  INVOLVING  THE  SAME 
TWO   UNKNOWN  ELEMENTS  WHEN  BOTH  EQUATIONS  ARE  QUADRATIC. 

Eliminate  one  of  the  unknown  elements  by  division;  solve  the 
resultant  hiquach'atlc  equation  for  the  other. 

Replace  this  element  by  each  of  the  four  roots  so  founds  in  the 
equation  formed  by  equating  to  zero  any  remainder  or  divisor  that 
contains  the  other  unknown  element  in  the  first  degree^  and  solve 
for  that  element. 

E.g.,  to  find  the  four  values  of  x,  y  from  the  pair  of  equations 
2a;?/ +  5/— 195  =  0,     3a^  —  4ic?/  — 7  =  0  :    then. 
2?/.a;4-(5y^- 195)1  ^x^-^y-x-l  |3a;- (23/ -  582) 
H 


Qy'X^  —  Sy-'X—\4:y 
6y. 0^4-3  (5/ -195)  a; 

-(23/ -585)  a? -142/ 
^ 

-22/(23?/--585)a;-28/ 
-22/(23?/^-585)a;-(5/-195)  (23^-585) 

(5/-195)(23/-585)  -  28 / 


332  EQUATIONS.  [Xi.  pr. 

Equate  this  last  remainder  to  zero  ; 

then   •.•  115^- 7438/+ 114075  =  0, 

2      ofc         4563 
.-.  V  =25  or   , 

^  115 

.-.  y  =±5  or    ±    ^^' 


V345' 
and  X  =±1  ov    T  J^-  [repl.  y 

This  process  consists  in  replacing  the  two  given  equations 
by  two  new  equations  got  from  the  last  two  remainders,  the  one 
free  from  a;,  and  the  other  having  x  only  in  the  first  degree. 

So,  to  find  ic,  ?/  from  the  pair  of  quadratic  equations 
x^-^2y-='^xy^    l(jx—\2y  =  bxy\ 
then   *.'  the  resultant  of  these  two  equations  is 

5?/*  —  14^^*  -\-Sy^=  0,  whose  linear  factors  are 
y^y^  (52/ -4)  (2/ -2), 
.-.  2/  =  0,  0,1,  2, 
and  a;=0,  6,  i,  4. 

The  locus  of  the  equation  0^4-2?/^  — 3a;2/  =  0,  i.e.,  of  the 
equation  (x  — 22/)  (aj  — 2/)=^0,  consists  of  the  two  straight 
lines  whose  equations  are    x  —  22/=0,    x  —  y  =  0. 

The  four  values  of  y  give  the  ordinates  of  the  four  points 
where  the  hyperbola  whose  equation  is  16a;— 12?/  =  5ic?/  cuts 
these  two  straight  lines,  and  two  of  these  four  intersections 
coincide  at  the  origin  ;  hence  the  double  solution,  a;  =  0,  ?/=  0. 

Note  1.     If  both  the  equations  be  of  the  form 

ax"  +  hxy  +  cy^-\-f=  0,    a^x"  +  h'xy  +  c^y''  +/'  =  0, 
the  following  method  may  be  adopted :    subtract  /  times  the 
second  equation  from  /'  times  the  first,  and  divide  the  resulting 
equation  by  y^ ;  solve  for  {x  :  y)  ;  .replace  a;  by  a  function  of  y, 
and  solve  for  y. 

More  generally  :  if  the  two  equations  consist  of  two  ?ith  degree 
homogeneous  functions  equated  to  constants :  eliminate  these 
constants  and  thus  obtain  a  homogeneous  nth  degree  function 
equated  to  zero.  Divide  this  equation  by  2/'*,  and  solve  the 
nth  degree  equation  involving  the  ratio  x :  y. 


13,  §  15.]  .SIMULTANEOUS   EQUATIONS.  833 

More  generally  still :  if  one  fnuction  be  of  the  mtli  degree, 
and  the  other  of  the  ?ith  degree,  divide  the  1.  c.  mlt.  of  m,  7i  by  each 
of  them,  raise  the  two  functions  to  the  powers  shown  by  the 
quotients,  and  put  them  equal  to  like  powers  of  the  constants, 
thus  making  both  equations  of  the  same  degree.  Solve  as  above. 

Note  2.     The  two  equations  may  be  written 

Po  +  PiH hP;«  =  0,     Qo  +  QiH f-Qu  =  0; 

wherein  Pq,  Qq  are  free  from  a?,  y ;  p^,  Qj  are  homogeneous  and 
of  the  first  degree ;  •••  p„,  Q„  are  homogeneous  and  of  the  de- 
grees w,  n. 

If  the  given  equations  be  so  incomplete  that  only  a  few  of  the 
expressions  Pq,  •••  p«,  •••  Qo,  •••  Qn  l^e  present,  it  is  often  best  to 
put  vx  for  y  in  both  equations,  eliminate  x,  and  get  the  values  of 
V,  and  then  of  x,  y.     [Or  else  put  vy  for  x  and  eliminate  y.'] 

By  thus  putting  vx  for  ?/,  the  equations  become 

Po+Ui-xH hu^-ic"'=0,   Qo+Vi-fliH |-v„-aj"=0, 

wherein    Ui,  •••  u^,  Vi'",  •••  v„,  are  known  quantities  or  functions 
of  V  whose  degrees  >  1,  2,  •••  m,    1,  2,  •••  n. 

This  method  is  similar  in  principle  to  that  of  Note  1. 

E.g. ^  In  the  example  of  Note  1,  Po  +  P2  =  0,  Qq  — Q2  =  0? 
wherein  Pq,  Qo,  Pa,  Q2=/, /',  ax" +  hxy  +  cy"^,  a' x^ +h' xy  +  c' y"^  \ 
Us,  Vs  =  a  +  &v  +  cv^i  o!  -f-  Vv  +  c' v^ ;  and  v  is  found  from  the 
quadratic  {cf-c'f)v'  +  {bf  -b'f)v  +  af-  a'f=  0. 

Note  3.  Sometimes  the  solution  of  a  pair  of  equations  may 
be  simplified  by  changing  the  unknown  elements :  notably 
bj'  making  use  of  the  following  relations,  connecting  the  sum 
of  two  numbers,  their  difference,  their  product,  the  difference  of 
their  squares,  the  sum  of  their  squares  : 

half  sum  +  half  difference      =  greater  number. 

half  sum  —  half  difference      =  less  number. 

product  sum       x  difference  =  difference  of  squares. 

sum  of  squares  -f-  twice  product       =  square  of  sum. 

sum  of  squares  —  twice  product       =  square  of  difference, 
(half  sum) 2  +  (half  difference)^  =  half  sum  of  squares, 

(half  sum)-  —  (half  difference)^  =  product. 


334  .    EQUATIONS.  [XI.  pr. 

When  each  equation  is  symmetric  as  to  x,?/,  it  is  commonly 
best  to  take  symmetric  functions  of  a;,  y  for  the  new  unknown 
elements. 

When  one  equation  is  symmetric  as  to  ic,  3/,  and  the  other  as 
to  a*,  —  ?/,  it  is  often  best  to  take  a-f-y,  x  —  y  for  the  new 
unknown  elements. 

Sometimes  equations  not  originally  thus  symmetric  may  be 
made  so  by  transformation. 

E.g.^  the  resultant  of  the  pair  of  equations 

3in/  — 4ic-42/  =  0,     ar^  +  ^/^  +  a-f?/ -  26  =  0     is 
92/^  -  15?/^ -  2422/2  _,_  624?/  -  416  =  0, 
which  is  not  easily  reduced  to  a  quadratic. 
But  put   {x-^yy^2xy  for  m?  -+-  t/^,   and  write  the  equations  : 
2^xy-4.{x  +  y)=0,    {x+yy-\-{x-iry)-^xy-2Q,=^0, 
then         x  +  y=Q>,   xy  =  8   or  x-^y  =  --^^,   xy  =  -^; 
and  the  four  values  of  a;,  y  ai*e  found  from  these  two  pairs 

of  equations,  each  consisting  of  a  simple  equation 
and  a  quadratic. 
So,  to  find  X,  y  from  the  pair  of  equations 
x^y  =  lxy,     a?  +  y^  =  ^xy', 
square  the  first  equation  and  subtract  from  the  second 
to  find  values  of  xy ;  join  each  of  these  equations 
with  the  first  equation  to  find  values  of  aj,  y. 
So,  in  the  equations   a;  +  2/  =  4,   x*  +  2/^  =  82, 
put  u-\-v^x^   u  —  v  =  y] 

then  {u-i-v)-{-(u  —  v)  =  A,     w=2; 

and  {u-{-vy-{-{u-vy  =  82,     u'^  +  6u^v^-\-v^=:U, 

...  v'-\-24:V--2o  =  0, 
.'.  v'=l    or    —25,   v=  ±1    or    ±6i. 
.'.  X  =3,   1,    2-\-oi,    2—5i; 
2/=l,   3,    2  —  01,    2-\-bi. 
So,  to  find  the  five  values  of  x,  y  from  the  pair  of  equations 
a;  +  2/  =  4,     a^  +  2/^  =  244; 


13,  §  15.]  SIMULTANEOUS  EQUATIONS.  335 

then    •.•    {u-i-v)-\-{u  —  v)  =  4,  (u  +  vy -j-(u -vy  =  2U, 

.-.  the  five  values  of  V  are     qo,+1,~1,     -\-i^S,     —i^S<, 

the  five  values  of  a;  are   +go, +3,+l,  2+i^3,  2— i^3, 

and  the  five  values  of  2/ are   "oc, +1,''"3,  2— ^V3,  2+1^3. 

Note  4.  The  meaning  of  these  infinite  solutions  may  be 
interpreted  as  follows : 

Consider  the  equation  a; +  2/ =  4  as  the  limiting  form  of  an 
equation  aj  +  6?/=4,  whose  coefficient  b  gradually  approaches 
unity  as  a  limit :  one  of  the  pairs  of  values  of  x,  y  grow  larger 
and  larger  without  bounds,  and  the  solution  is  either  cc=+oo, 
y=-cc,  or  a;  =  ~oo,  ?/=+oo,  according  as  6  is  a  little  less  than" 
unity  or  a  little  greater. 

Note  5.    Sometimes  the  roots  of  higher  equations  may  be 
found  by  the  method  of  division. 
E.g.,  of  the  pair  of  equations 

y{x^  ^-r)  =  4(ic  +  2/)^     xy  =  4:{x  +  y), 
the  resultant  is    y*  —  8y^  =  0; 
and     •.*  this  function  of  y  is  divisible  by  y^, 

.'.  the  equation  has  three  roots  0  ;  it  has  also  one  root  8. 
But     *.•  the  general  resultant  of  a  cubic  and  a  quadratic  equa- 
tion is  of  the  sixth  degree, 
.'.  this  resultant  has  lost  its  two  highest  terms,  and  the 

equation  has  two  roots  oo  ; 
.-.  the  values  of  y  are  oo,  go,  0,  0,  0,  8, 
and  the  six  corresponding  values  of  x,  found  from   the 

equation  x  =  4:y:(y  —  4)  are  4,4,0,0,0,8. 
The  geometrical  interpretation  of  these  roots  is,  that  of  the 
six  points  of  intersection  of  the  loci  that  represent  the  two  equa- 
tions two  are  at  an  infinite  distance  and  lie  on  the  line  a;  =  4, 
three  are  at  the  origin,  and  one  is  at  the  point  whose  co-ordi- 
nates are  8,8;  or,  in  the  language  of  limits,  if  one  of  the  curves 


836  EQUATIONS.  [XI.  pr. 

change  its  form  slightly,  by  the  gradual  change  (say)  of  a  sin- 
gle coefficient,  and  thus  approach  its  present  form,  tlieu  two  of 
the  points  of  intersection  recede  to  an  infinite  distance,  three  of 
them  not  coincident  approach  the  origin,  and  one  approaches 
the  point  8,  8. 

So,  of  the  pair  of  equations 

a^+y(xy-l)  =  0,    f-x{xy-\-l)  =  0, 
the  resultant,  found  from  the  last  remainder,  is 

4y^-.4:f-y=0, 
and  the  second  last  remainder  gives 
{2y'-{-l)x-2f==0, 
.-.  2/  =  0    or    2^  =  i(l±V2); 
i.e.,         y  may  have  the  value  0  or  any  one  of  the  eight  values 

•    of  ■</*(!  ±v2),  =</a±vi). 

and  ic,  =22/^:(22^-}-l),  may  have  the  value  0  or  any  one  of 

the  eight  values  of  ^1  :  (2  ±  2  V2),  =  -^(-i-±  Vi)- 
Note  6.  Special  methods  of  solution  :  Many  sets  of  simul- 
taneous equations  may  be  solved  by  special  devices.  The  ex- 
amples given  below  are  meant  merely  as  suggestions  to  the  reader. 
He  is  advised  to  tr}-  his  own  ingenuity  upon  each  example  before 
studying  the  solution  here  shown  ;  and  afterward,  to  see  how 
far  the  principle  of  each  solution  applies  to  other  examples. 

1.    To  find  the  values  of  x,  y  from  the  pair  of  equations 
a^  +  a;-y  +  2/4  =133,   (1)     x'-xy -\-y-=l :  (2) 


then 

x^-\-xy  +  f    =19,     (3) 

[div.(l)by(2) 

and 

ar^+2/'  =  13;                (4) 

[add  (2),  (3) 

.•.  xy         =6,                   (5) 

[sub.  (4)  from  (3) 

a:2  +  2x?/  +  2/'  =  25,     (6) 

[add  (3),  (5) 

and 

x^-2xy  +  f  =  l',      (7) 
.•.x  +  y  =  ±b,     x  —  y=z±l', 

[sub.  (5)  from  (2) 

.-.  0^,2/ =+3, +2;  -3,-2;  +2,-^3; 

-2, 

-3. 

2. 

If  x^'-x'  +  y^-y-^S^,    x'^x^y'^^ 

2/2=49: 

then 

(a^  +  rY  -  2  x'f  -  (x'  +  ?/-)  = 

=  84, 

(a^  +  /)+a^2/'  =  49. 


13,  §  15.]  SIMULTANEOUS  EQUATIONS.  837 

Put  u  ■■=  0?  -\- if  ^  V  =  X- y"^  \  (1 ) 

then   •.•  w^-u- 2^=84,    ?^  +  v=49;  (2) 

.-.  U-  +  u  =  182  ;  [add  (1)  and  twice  (2) 

.-.  u  =  n   or    -14. 

(1)  Puta^  +  2/'=13; 

then   •••  x-if  =  ZQ,   cc^ 4.363,-2^13. 

.-.  oir=      9   or       4,    i/=z      4   or       9; 

.-.  X  =  ±3    or    ±2,    ij  =±2   or    ±3; 

i.e.,a;,2/=-^3,+2;+3,-2;-3,+2;-3,-2;+2,+3;+2,-3;-2+3;-2,-3; 
eight  pairs  of  roots. 

(2)  Futsc^-{-y^  =  -U; 

then         a^f-  =  G3,    x' -{-6Sx-^=:-U', 

.-.  x'  =  —  7±i^U,  2/^=  — 7q:iV14; 

.-.  X  =±V(-7±*V14),    y  =±v(-7:fiV14); 

eight  pairs  of  roots. 
The  plats  of  these  two  equations  intersect  in  only  eight  real 
points  ;  the  other  eight  points  of  intersection  are  imaginary. 

3.  If  V(^+2/)4-v/(^-^)  =  V«.  V(^'+2/')+V(^-2/')=^  = 
then  2a;+2V(aJ^-/)=a,  2a^+2^{itl^-y^)=b'',  [sqr. 
and          oif —  y-  =  ia^ —  ax-\-a:^,       x^  —  y^  —  \h^  —  h^a?-\-Qi^^ 

[div.  by  2,  transp.,  sqr. 
I.e.,  y'^  =  ax  —  ^a?^  y^z=  ly^y?  —  \h^ ; 

.♦.  'b^x^-\y'=^{ax-\ay\ 
I.e.,  (a2-Z>2)^_ia3.^4.(_i^^44.j54)^0; 

.-.  a;  =  [a-^±6(a2-262)]:4,(a2-62), 

2/2=  {ax  -  \a-)  =  ah  •  \_ah  ±  (a-  -'26^)] :  4(a2  -  ft^)  ; 

2/  =  ±  V[a6  .  (a6  ±  a^  ip  26^) :  A{a^-h'~)^. 

4.  If  a;(a;+^+2;)=18,    2/(a^+2/+2;)=  12,    z{x+y^z)  =  Q'. 
then  (i»-i-?/  +  2;)(aJ  +  2/  +  2;)  =  36,  [add 

.-.  x  +  y^z  =±6, 

.-.  fl7=±3,     2/  =  ±2,    z=±l,  [div. 


338  EQUATIONS.  [XI.  pr. 

5.    If  xyz  =  a-{y  -\-z)  =  h-{z  +  x)  =  c-{x  +  ?/)  : 

\zx     xy)         \xy     yzj         \yz^zxj 


then 


T^^  111 

Put  u,  V,  w  =  — ,   — ,   — ; 
yz     zx     xy 


then   •••  l  =  a\v-j-to)  =  b\iv-\-u)  =  c^(u  +  v), 


1/1,1      i\ 

.-.  x^,=  —  :f Y  =ic:(2v.v) 

yz    \zx   xyj 

So  for  2/^,  2^ 

6.   If  yz-^zx  +  xy  =  26,  (1) 

yz(y-{-z)-\-zx{z-\-x)-}'Xy{x  +  y)=162,        (2) 
yz  {f  +22)  +  zxiz-"  ■^x')-\-xy  (^2+  /)  =  538 :  (3) 
Write  (2)  in  the  form 

yz{x+y-\-z)+zx{x^y-\-z)-Jrxy{x^y+z)—^xyz=lQ>2, 
then  {yz-\-zx-^xy){x  +  y -{-z)  —  ^  xyz  =10,2, 

and  26 (a; +  ?/+ 2) -3 3^2/2  =  162.  (4)   [(1) 

So  write  (3)  in  the  form 

{yz+zx+xy){x^-\-y'^+z')  —xyz{x-\-y+z)  =538; 
then  2^\_{x-\-y-\-zY-2{yz-\-zx  +  xy)'] 

-xyz{x-\-y  +  z)  =  538.  (5) 

Put  w,  V  =x  +  y  -\-z^   xyz  ; 
then         26it-3v=lG2,    26^^- wv=  1890,        [sub.in(4,5) 

.-.  w  =  9    or  -  12^\,     ^  =  24    or    -159, 
i.e.,  x-^y  +  z^d   or    —12-^,    xyz  =24:    or    —159; 


13,  §15.]  SIMULTA]SEOUS   EQUATIONS.  339 

but     •.•  x{y+z)-^yz=:2Q>,  ,  [(1) 

.•.  cc (9  —  x)  +  24fl;~^  =  26,  [use  first  vals.  for  w,  v 

i.e. ,  a^  -  9  a^  +  26  a;  -  24  =  0, 

or  (if- 2)  (a;- 3)  (x -4)  =  0;  "[factor 

.-.  a;=2,  3,  4;  [th.2cr.l 

and  2/=2,  3,  4,    2;  =  2,  3,  4.  [symmetry 

These  roots  may  be  grouped  in  six  different  ways : 
a;  =  2,     2,     3,     3,     4,     4; 
2/ =3,     4,     4,     2,     2,     3; 
2=4,     3,     2,     4,     3,     2. 
So,  for  the  sets  of  values  from  the  other  values  of  w,  v. 

7.   If  2/2+  2/2+^2^7,  (1) 

s2+2a;  +  aj-=13,  (2) 

a^  +  a^2/+2/'  =  3:  (3) 

then  (a;-?/)  (a; +  2/ +  2)  =  6,    (2 -2/)  (a:  +  2/ +2;)=  10, 

[sub.(l,  3)fr.  (2) 
.*.  x  —  y:  z  —  y=  3  :  5, 

i.e.,  bx  —  by  —  2>z  —  2>y   and    2?/  +  32;  =  5a;;        (4) 

but     •••  22  +  a;(2!  +  aj)  =  13,  [(2) 

...  22_pj(22/  +  3;z).-J-(22/4-8;2)=13,       [sub.  for  a;  fr. (4) 

I.e.,  42/2  + 222/2; +  492;2  =  325,  (5) 

•••  2/=±2,  ±-1-;    2  =  if3,  ±^;    a^=  +  l,  ±     ^ 


8.    If  a;2_2/2  =  a  (1),  y--zx  =  h  (2),  2--aJ2/  =  c  (3)  : 
From  the  square  of  one  subtract  the  product  of  the  other  two  ; 
then         a?  (ar^  +  2/^  +  ^r^  —  3  xyz)  =  a^  —  &c  =  a, 
y  (ar^  +  7/  -\-z^  —  2)  xyz)  =  b^  —  cc(  =  b, 
and  z  {x^  -\- y^  -{- z"^  —  S xyz)  =  c^  —  a6  =  c. 

.  • .  A2/  =  Bx,    Az  =  ca; ; 
.-.   (a^  —  ■Bc)o(y^  =  aA^',  [sub.  for 2/,  2;  in  (1) 

_       yg  •  A g^  —  &c 

*'*  ^~  -^{a^-bc)~  ^{d'-\-b^-\-c'-3abc)' 
c  W  —  ca  &  —  ah 


V(a'+^'+c'-3g6c)'  V(«'+^'+c''-3a6c) 


340  EQUATIONS.  [XI.  pr. 

§16.    SPECIAL  PROBLEMS   INVOLVING  QUADRATICS. 

For  definition  of  special  problems  and  for  the  method  of 
putting  such  problems  into  equation  and  of  discusbing  the  solu- 
tions, see  §  10.     These  methods  are  best  shown  by  examples. 

1.  Two  farmers  at  a  fair  each  spent  SHOO,    a  bought  50  sheep 

and  12  cows  ;  b  bought  50  more  sheep  than  cows  ;  and 
the  sum  spent  by  the  two  together  in  the  purchase  of 
sheep  was  half  the  joint  expenditure.  What  was  the 
price  of  cows  and  what  the  price  of  sheep,  and  how 
many  sheep  and  how  many  cows  did  b  buy  ? 
Let  tt,  v,  x,  y  =  the  price  of  the  sheep,  the  price  of  cows,  the 
number  of  sheep,  and  the  number  of  cows  bought  by  b  ; 
then  •.•  50^-1-121;=  1100,  (1)  xu+yv=  1100,  (2) 
50w  +  a;w=1100,      (3)         a;  =  ?/  +  50;  (4) 

.-.   (a;— 50)w  +  (y-12)i;  =  0,   xu  —  12v  =  0, 

[sub.(l)fr.(2)  andfr.(3) 
.'.  yu-{-{y-12)v  =  0,    {y -\-50)u-12v  =  0, 

[sub.  fora;fr.(4) 
.-.   [122/4-(2/  +  50)(7/-12)]<^  =  0.  [elim.^ 

Put       12?/+ (2/  + 50)  (2/- 12)  =  0;  [th.  2  cr.  1 

then         ^=10  or  -60. 

Reject  the  root  —  60  as  absurd ; 
then         2/=10;    ic,  =.vH-50,  =  60  ; 

.-.  50?i  + 12^=1100,    60w  +  10'U=1100, 

[sub.  for  a;,  ^/in  (1,  2) 
.-.  u=10,    -^  =  50; 
i.e.,  b  bought  60  sheep  at  $10,  and  10  cows  at  $50. 

2.  The  fore-wheel  of  a  carriage  makes  6  revolutions  more  than 

the  hind-wheel  in  going  120  yards  ;  but  if  the  circumfer- 
ence of  each  wheel  be  increased  one  yard,  the  fore-wheel 
will  make  only  4  revolutions  more  than  the  hind-wheel 
in  the  same  space ;  find  the  circumference  of  each  wheel. 
Let  ic,  ?/  =  circumferences  of  fore-wheel  and  hind-wheel ; 

,,  120      .      120       120        .        120  ,  ,  . 

then 6  = , 4  = ;   and  07  =  4,  y  =  o. 

X  y       x  +  1  2/  +  1 


14,  §17.]  BINOMIAL   EQUATIONS.  Cil 

§17.      BINOMIAL    EQUATIONS. 
PrOB.  14.     To    SOLVE    AN   EQUATION    OF   THE    FORM    X'^  =  a". 

Transpose  a",  giving  x°  —  a"  =  0  ;  factor  x"—  a",  and  pvt  tJie 
factors  severally  equal  to  0  ;  solve  the  equations  thus  formed. 
The  roots  of  these  new  equations  are  the  roots  sought.  [ih.AcY.1 

1 .  To  solve  the  equation   x-  =  a^: 
then    • .'  x^  —  a^  =  {x  —  a) ' (x  +  a) , 

.'.  x  —  a  =  0,    or  a;  +  a  =  0, 
.♦.  x  =  a,    or    —a. 

2.  To  solve  the  equation   x^=  —  a^: 

then   •.*  x^ -{-a^  =  (x  —  a-^—l)-{x-^a^—l), 

[VIII.th.2,df.imag. 
.-.  x  =  a-y/—l^    or    — a^— 1, 

I.e.,  x  =  ai,    or    —ai. 

3.  To  solve  the  equation   (ii^  =  a^: 
then    •.•  x^ —  a^  =  {x  —  a)'{x^ -^ax -^-a^), 

.-.  a;  — a  =  0,    or  a^  +  Ga;  +  a^  =  0, 

.'.  x  =  a,   or  |^a(  — l±iV3).  [sol. quad. 

4.  To  solve  the  equation   a^  =  —  a^: 

then    *.•  the  equation  £c^  =  — a^  gives  a^=a^  if  —a;  replace  a;, 
.-.  x  =  —  a,    or   -^a(l  ±  i-y/S). 

5.  To  solve  the  equation   x'^  =  a'^: 
then    •.•  X* -  a^  =  {3^ - a')'(x'  +  a^), 

.'.  x  =  a,    —a,    m,    —a,i.  [1,2 

6.  To  solve  the  equation   x*  =  —  a* : 

then    '.'  x'  +  2a'x--\-a*=^2a'a^,       .  [add  2a2a^ 

r.e.,  (a;2  +  a2)2-2a^a;2  =  0; 

...  (a;2  ^  ^^2  _  ^^^  ^2) .  (a^  +  a2  +  aaj  V^)  =  0, 

.-.  a;  =  |a(-V2±^V2),    }a(V2±*V2)-     [sol.  quad. 


342  EQUATIONS.  [XI.  pr.  U. 

7.  To  solve  the  equation   a.*^  =  a* : 

then    *.•  ic*  —  a*  =  (a;  —  a)  •  (a;*  +  if^a  +  ar^a^ -I- ica^  +  a*) , 
.'.  x  =  a, 

and  a;*4-a^a4-ar^a"  +  a;a^  +  a*  =  0 ; 

.*.  ay^ -\-ax-\-a'-\-a^x~^  |-a*a;~-  =  0,  [div.  by  ar* 

.-.  (a^  +  a'a;-')  +  a(a;  +  a^a;-^)  +  a'  =  0, 

.-.  {x  +  a-x-Y+  a{x-}- a-x-')  -2a^-{-a^  =  0, 

.'.  a;  +  a2a;-i  =  Ja(-l±V5), 

.-,  x  =  a,   ^a[(V5-l)±iV(10  +  2V5)], 

-ia[(V5  +  l)±iV(10-2V5)].  [sol.  quad. 

8.  To  solve  the  equation   T'  =  —  a^: 

then    *.•  the  equation   v?  =  —  c^  gives  ^  =  o^  if  —a;  replace  a;, 
.-.  x^-a,    -Ja[(V5-l)±*V(10  +  2V5)], 

9.  To  solve  the  equation   a^  =  a^: 
then    •••  a^-a«  =  (af'-a-^).(ar^H-a^), 

,',  x  =  a,   |a(-l±tV3)'    -»'   Ja(l±iV3).      [3,4 

10.  To  solve  the  equation   x^  =  —  a^: 

then    *.*  the  equation  oif  =  —  a^  gives  x^=:a^  if  ?a;  replace  a;, 
.-.  x  =  ai,    ^a{—i±  ^3),    —ai^   ia(i±  ^3). 
And  so  on  for  other  roots. 

Note.   Another  method  of  soltition  is  shown  in  X.  Prob.  1,  in 
finding  the  nth  roots  of  a**  and  of  —  a". 

§18.     LOGARITHMIC   AND   EXPONENTIAL   EQUATIONS. 

The  methods  of  solving  such  equations  are  set  forth  in  IX. 
Probs.  3,  8. 

J57.gr.,    to  find  a;  from  the  equation    15^*  +  6  •  15*  =  51975  : 
then   •.•  15^=225    or    -231,  [sol.quad. 

.-.  a;  =  log  225  :  log  15  =  2.3522  :  1.1761  =  2  ; 
but  of  the  equation   15"'=  —  231  no  solution  is  possible. 


§  19.]  EXAMPLES.  343 


§  19.      EXAMPLES. 

§1. 

1.  From  the  following  statements  pick  out  the  sufficient  condi- 

tions, the  necessary  conditions,  the  equivalent  statements, 
the  associated  statements,  the  incompatible  statements, 
the  independent  statements : 
ic<6,  a;=3,  x<D,x>4:,  2x=6,  fl^  =  9,  x  =  3,  Xi^S. 

2.  Give  examples  in  which  one  statement  is  a  ^      ^  .  '  ^'  but 

not  a  ^  sufficient,    (.(j^^iiti^^n  ^f  another. 
'  necessary, 

3.  Show  that  if  one  statement  -{  ,  ^      ,  a  <j  g,j«^^-^%^  condition 

of  another,  then  the  latter  -J  .        .  a  <(  condition 

'  '  IS  not      '  necessary 

of  the  former ;  give  examples  of  these  four  cases. 

§  5,  PROB.  1. 

•  ••  24.    Solve  the  equations  : 

4.  12  — 5a;  =  13-a;;    l-ox=7x+3;    6a;-5(3a;-7) -21  =0. 

5.  a  —  2x  =  x  —  b',    7n  —  nx=px-\-q;    ax  —  b{x  —  l)  —  c  =  0. 

6.  {x+l){x-l)  =  x(x-2);   (x-\-4){x-2)  =  (x-d){x-S), 

7.  (x-\-a)(x—b)  =  (x—c){x-{-d);    {x--m)(x-\-n)=x(x—q). 

8.  ax  —  m  —  2\bx  —  n  —  S  [^cx —p  —  4:(dx  — q)']\  =  0. 

9     Q,      4  — a;_ll.   lx-\-A:.     5  — a;_22      x_^  —  lx 
S^S'O  a~3       2      ~^ 

10.  l(x-\)-^{x-2)  =  ~',    l(oa;-6)--(a;-l)  =  a;-2. 
a  b  b      m  n 

11.  l(a;-t-l0)-|(3a^-4)  +  i(3aj-2)(2a;-3)  =  a^--|. 
o  o  0  lo 

12       1_A4_A=:A.        -1—4         ^  ^ 


13. 
14. 


X      2x      7x      28       x-\-l       x-{-2      x  +  3 
2a;  — 3_  6x  +  5         x  —  a  _3x  —  c 
3a;  +  4~~9a;  — 10'     2x  —  b'~6x-d 

_[. 1_  __1 1_  .  x—1      x—2_x—Z     a;-4 

x  —  Z      x  —  4:      x  —  b     x  —  Q^  x—2      x—3      x—4:     x—5 


344  EQUATIONS.  £XI. 

2\      3j     S\      4y     4V       5y  x-a     x-b      x-c 

16.  {x-ay-^{x-by-\-{x-cy=S{x-a){x-b)(x-c). 

17.  {x'-'Sx  +  4.)^==x-S;  [2(l-a;)(3-2a;)]-^  =  2a?H-l. 

18.  (8~4x)^  +  (13-4a;)*=5;    2x+^l4.x'-\-^(l-4rX)}  =  l, 

19.  18:  V(2a;  +  3)  =  V(2aJ-3)4-V(2a;  +  3). 

20.  3V(aJ-|)  +  7V(^  +  A)  =  10V{aJ  +  T*ir)- 

21.  v(3a;H-l)-V[2-a;  +  2V(l-a;)]  =  l. 

22.  -^/(V3+a;v7)+^(V3-a;V7)=-^12. 
23  l+x-V(2a;  +  ar^)^^V(2  +  a;)-Var 

24.  V(«  +  a;)H-V(a-a5)  =  &[V(a  +  »)-V(«-^)]- 

§§  6,  7,  PROBS.  2,  3,  4. 

•  ••28.  From  the  following  pairs  of  equations  eliminate  one  of  the 

variables  by  the  first  method,  and  solve  the  equations  : 

25.  8x  +  Sy=U,   5^=10;    3a;-8y=7,   3^x  =  5, 

26.  15x  +  22/=17,    9x  —  4y=5. 

27.  210x  +  42y  +  93  =  0,    22a;  + 14y  +  7'=  0. 

28.  |2/-ia^  +  24  =  0,   ^2/  +  ia;-f- 11  =  0. 

•  ••31.    So,  by  the  second  method  : 

29.  x  +  y  =  9,   a;-2^=l;     5x-{-Sy  =  8,    7x-Sy  =  4:. 

30.  3a;  +  2/=16,    32/4-a;=8;    Sy  =  5x,    16y  =  27x-l. 


y)' 


3V4      5      6;      4^        ^^'     2V4      5      3;      4^    ^ 
•  •35.    So,  by  the  third  method  : 

32.  l\x-3y  =  0,   x-y  =  -U',     x  =  iy,   x-^y  =  ^, 

33.  x-y  =  i,    x+i=i(y  +  x). 

34    _£±^=3      a;-3.v      5y-a;^l 
'    a;-22/        '  6  9  2 

35.    ^(804-3a;)  =  18i-i(4^  +  32/-8), 
102/  +  i(6^  —  35)  =  55  +  10a;. 


§  19.]  EXAMPLES.  345 

•  ••39.    Eliminate  x  by  the  fourth  method  from  : 

36.  4a;-7?/  =  9,    IGo^^- 49/=  207. 

37.  if^  +  £c?/=7,    ic4-?/  =  4;    a;^  +  3a7?/ =  10,    2/^  +  2ic?/  =  5. 

38.  Q^-\-xy^f  =  h,    2a^  +  3a;?/  +  42/-  =  ll. 

39.  a;^  +  a^2/  +  a^?/2  +  »/  + 2/4  =  1,   oj^-f- 2/^  =  2. 

§  7,   PKOB.  5. 

•  ••  48.    Solve  the  systems  of  equations  : 

40.  3a;  — 4?/ +  52;  =4,   ^x  —  y  —  z  =  ^^    7x  —  5y—Sz  =  —l. 

41.  a;  +  2/  +  2  =  6,   a;  — 2/  +  2f=2,   x  +  y  —  z  =  0. 

42.  a;— 2?/— 52;=20,    3a;— 5?/— 32=22,    — 8a;+ll2/+92=  — 57. 

43.  2(a;+l)-3(?/-l)  +  2-2  =  2, 

2(a;-l)  +  4(2/  +  l)-5(2-l)  =  3, 
3(2aj  +  2)-2(2/-l)4-3(2+l)=29. 

X     y     z       6^   X     y      3z      18     x     y      7z      21 

45.  3Ja;+5|2/-lTT2'=51,  2/^  +  22  =  2 a^+^ajS^  c?/  +  62  =  a, 
2ia;+3i2/-lj2;  =23f,  z'' -haP=2b^ -j-^y^  az+cx  =  b, 
lia;+2i2/4-  f^^  =31|H;  a;2+2/'  =  20^ +|-2;2;  6a;+a2/  =  c. 

46.  a;  +  22/+32!  +  4w  =20,   x-^2y -\-Sz-4:U  =12, 
x-^2y  —  Sz-t4u  =8,     a;- 2?/H-32 +  4z«  =  8. 

47.  3a;  — 4?/  +  32;  +  3v-6w=ll, 

Sx  —  5y  +  2z  — 4:11=11,     lOy -3z -{-Su  —  2v  =  2, 
5z+Au-[-2v-2x  =  3,         6w- 3v  +  4a;- 2?/=  6. 

48.  5a;-2(2/  +  2;  +  'u)  =  -l,     -  122/  +  3  (2  4-v  +  a;)  =  3, 

42  — 3  (v  + a; +  2/)  =  2,     8^  —  (a; +2/  +  ^)  =  —  2. 

Denote  x  +  y-i-z-j-vhys;  from  these  equations  respectively 
express  a?,  2/,  2,  v  in  terms  of  s ;  substitute  these  values  in  any 
one  of  the  equations  ;  solve  for  s ;   and  thence  find  x,  y,  z,  v. 

§8. 

•  ••  52.   Plat  the  lines  that  represent  the  equations  : 

49.  a;  =  0,   y  =  0,   a;  =  4,   a;=— 4,   2/  =  4,   2/=— 4,  x=±a, 

y=  ±b,  [a,  6,  lines  of  any  known  length 

50.  y  =  x,   y=  —X,  y  =  2x,  y=  —  Sx,  2y  =  Sx,  3y=—2x, 

ly  =  mx,   ly=—  mx,  [Z,  m  any  two  given  numbers 


34G  EQUATIONS.  [XL 

51.  y  =  x-}-2,   y=-x-{-2,   y  =  x-2,   y=-x-2, 

y  z=  mx  -\-  c,  [_m  an}'  given  number,  c  any  given  line 

52.  2a;  +  3  2/  +  5  =  0,    3  a;  — 2?/  — 5  =  0,    Ix -^-my -{-c  =  0, 

[Z,  m  any  given  numbers,  c  any  given  line 

53.  Find  the  lengths  of  the  intercepts  upon  the  axes  of  the 

lines  whose  equations  are  given  in  Exs.  52-55. 

54.  Find  the  co-ordinates  of  the  points  of  intersection  of  the 

lines  whose  equations  are 

y  =  x,   y=—x;  y  =  x-\-2,   y  =  x  —  2; 

2x4- 3y +  5  =  0,  3a;  — 2?/  — 5  =  0;    y  =  mx+c^ 

y=  —  wia;  +  c,  [m  an}'  given  number,  c  any  given  line 

55.  Find   the  co-ordinates   of  the    vertices   of    the   triangles 

bounded  by  the  lines  that  represent  the  equations  : 

2a;-J- 37/  +  5  =  0,    3a;  — 2?/  — 5  =  0,   a;  =  5; 

ax-^by  +  c=0,   a'x  +  b'y+c' =  0,   a"x-{-b"y -\-c"  =  0. 

56.  Find  the  co-ordinates  of  the  vertices  of  the  parallelograms 

bounded  by  the  lines  that  represent  the  equations  : 
2a;-f 32/-f5  =  0,    2a;-f 3?^- 5  =  0, 
3a;  — 2?/  +  5  =  0,    Sx-2y-5  =  0, 

57.  By  aid  of  Bezout's  method  solve  examples  42-51. 

§  10,   PROB.  6. 

58.  Find  two  numbers,  such  that  their  sum  is  27 ;   and  that, 

if  four  times  the  first  be  added  to  three  times  the  second, 
the  sum  is  93. 

59.  Find  two  numbers,    such  that  twice  the   first  and  three 

times  the  second  together  make  18 ;  and  if  double  the 
second  be  taken  from  five  times  the  first,  7  remains. 

60.  A  flagstaff  is  sunk  in  the  ground  one-sixth  part  of   its 

height,  the  flag  occupies  6  feet,  and  the  remainder  of 
the  staff  is  three-quarters  of  its  whole  length ;  what  is 
the  height  of  the  flagstaff? 

61.  The  diameter  of  a  five-franc  piece  is  37  millimeters,  and 

of  a  two-franc  piece  is  27  millimeters  ;  thirty  pieces  laid 
in  contact  in  a  straight  line  measure  one  meter;  how 
many  of  each  kind  are  there  ? 


§  19.]  EXA3IPLES.  347 

62.  Find  three  numbers   such   that  the  sum  of  the  first  and 

second  is  15  ;  of  the  fii'st  and  third,  16  ;  and  of  the  second 
and  third,  17. 

63.  The  sum  of  the  three  digits  of  which  a  number  consists 

is  9  ;  the  first  digit  is  one-eighth  of  the  number  consist- 
ing of  the  last  two,  and  the  last  digit  is  likewise  one- 
eighth  of  the  number  consisting  of  the  first  two. 

64.  At   an  examination  there  were   17  candidates,    of  whom 

some  were  passed,  some  conditioned,  and  the  rest  re- 
jected; if  one  less  had  been  rejected,  and  one  less 
conditioned,  the  number  of  those  passed  would  have 
been  twice  those  rejected,  and  five  times  those  condi- 
tioned ;  how  many  of  each  class  were  there  ? 

65.  There  are  three  candidates  at  an  election,  at  which  it  is 

necessary  that  at  least  one  more  than  half  the  entire 
number  of  electors  should  vote  for  the  successful  can- 
didate ;  A  fails  to  obtain  an  absolute  majority,  although 
he  has  20  votes  more  than  b  ;  but  supposing  that  c,  whose 
votes  are  only  three-tenths  of  b's,  had  withdrawn,  and 
that  one-fourth  of  his  supporters  voted  for  a,  then  a 
would  have  been  barely  successful ;  how  many  voted  for 
each  candidate  ? 

66.  A  gentleman  left  a  sum  of  money  to  be  divided  among 

four  servants  ;  the  first  was  to  have  half  as  much  as  the 
other  three  together,  the  second  one-third  as  much  as  the 
other  three,  and  the  third  one-fourth  as  much  as  the 
other  three ;  the  first,  moreover,  was  to  have  $  70  more 
than  the  last ;  how  much  should  each  get  ? 

67.  A  father  divides  his  estate  among  his  children  as  follows  : 

to  the  first  a  dollars  and  the  nth.  part  of  the  remainder ; 
to  the  second,  2  a  dollars  and  the  nth  part  of  the  remain- 
der ;  to  the  third,  3  a  dollars  and  the  ?ith  part  of  the 
remainder ;  and  so  on.  It  results  that  in  the  entire 
division  of  the  estate  each  child  receives  the  same 
amount.  Find  the  value  of  the  estate,  the  number  of 
children,  and  the  amount  each  one  receives. 


348  EQUATIONS.  [XI. 

68.  In  a  company  of  a  persons  each  man  gave  m  dollars  to  the 

poor,  and  each  woman  n  dollars ;  the  whole  amount 
collected  was  ka  dollars  ;  how  many  men  were  there,  and 
how  many  women  ? 

Show  that,  if  m>n,  then  m>k>n. 

Show  that  the  example  is  possible  only  when  (m — k)  a,{k—n)a 
are  multiples  of  m  —  ?i  and  have  the  same  sign  as  m— ?i. 

69.  Upon  a  horizontal  straight  line  let  o  be  a  fixed  point,  let  a 

lie  a  units  to  the  left  of  o,  and  b,  5  units  to  the  right  of 
o  ;  fmd  on  this  line  a  third  point  x  such  that  if  m  be  the 
middle  of  bx,  then  ao  is  one-third  of  am. 
Show  that  if  4  a  >  6,  x  lies  to  the  left  of  o  ;   if  4  a  =  6,  x 
coincides  with  o  ;  if  4  a  <  &,  x  lies  to  the  right  of  o. 

70.  A  reservoir  holding  v  gallons  is  filled  in  h  hours  by  p  pipes, 

all  of  the  same  size,  and  by  the  rain  falling  uniformly  on 
a  roof  of  8  square  yards.  Another  reservoir  holding  v' 
gallons  is  filled  in  h'  hours  by  p'  pipes  of  the  same  size 
as  the  others,  and  the  rain  falling  uniformly  and  with  the 
same  intensity  as  before  upon  a  roof  of  s'  square  yards. 
Find  X,  the  inflow  per  hour  of  each  pipe,  and  y,  the  rain- 
fall per  hour  on  each  square  yard  of  roof. 
Explain  the  meaning  of  the  jjroblem  if  for  particular  values 
of  the  constants  either  x  or  y  or  both  of  them  be  negative. 

71 .  Two  circles  of  radii  r,  r'  lie  in  the  same  plane  and  have 

their  centres  d  units  apart ;  find  the  point  where  the  exte- 
rior common  tangents  cut  the  line  that  joins  the  centers. 
Show  by  the  formula  that  if  the  smaller  circle  grows  while 
the  larger  stands  fast,  the  point  recedes  farther  and  far- 
ther away ;  that  when  the  growing  circle  is  of  the  same 
size  as  the  other,  that  point  has  gone  to  infinity  (does 
not  exist)  ;  and  that  when  the  growing  circle  passes  the 
other,  the  point  reappears  upon  the  other  side  at  infinity, 
and  creeps  back  toward  the  circles. 

72.  Find  the  four  terms  of  a  proportion  that  exceed  by  the 

same  number  the  four  numbers  a,  6,  c,  d. 
Discuss  the  solution  when  (1)  ad  =  bc,  (2)  a-\-d  =  b  +  c. 


§  19.]  EXAMPLES.  349 

73.  Given    the    series    a +  5,    aj)-\-bq^     ap^-\-bq^^    ajy^-j-bq^, 

ap^  4-  bq^f  •••,  to  find  two  numbers  x,  y,  such  that  each  term 
of  this  series  after  the  second  can  be  got  by  multiplying 
the  one  before  it  by  a;,  and  the  one  before  that  by  ?/,  and 
adding  the  products. 

74.  Given  the  series  a  -f-  5  -j-  c,  ap-\-bq-{-  cr,  ap^  -j-  bq^  -j-  cr^,  •  •  • , 

to  find  three  numbers  x,  y,  z^  such  that  each  terra  of  this 
series  after  the  third  may  be  found  b}^  multiplying  the 
one  before  it  by  x^  the  one  before  that  by  ?/,  and  the  one 
before  that  by  2,  and  adding  the  products. 

75.  A  laborer  receives  a  dollars  a  day  when   he  works,  and 

forfeits  b  dollars  a  day  when  idle.    At  the  end  of  m  days 

he  receives  h  dollars  ;  how  many  days  does  he  work,  and 

how  many  is  he  idle? 
What  relation  exists  between  the   given  elements   if   his 

forfeits  just  cancel  his  earnings?  if  his  forfeits  exceed 

his  earnings? 
Give  numerical  illustrations. 

76.  A  father  is  now  a  times  as  old  as  his  son  ;  h  3ears  hence  he 

will  be  b  times  as  old ;  what  are  their  ages  now  ? 
Give  numerical  values  to  a,  &,  A:,  and  interpret  the  results. 
Show  that :  7v  >  0  if  a  >  5  ;  ^'  =  0  if  a  =  6  ;  h<0\i  a<b. 

77.  The  sum  of  two  numbers  is  a,  and  the  difference  of  their 

squares  is  k'^ ;  what  are  the  numbers  ? 
Interprettheresultsif  (1)  k->a-\  (2)  k^  =  a^;  (3)  k^<aK 

78.  The  difference  of  two  numbers  is  a,  and  the  difference  of 

their  squares  is  k^ ;  what  are  the  numbers  ? 
Interpret  the  results  if  (1)  k^  >  a^ ;  (2)  k'  =  a' ;   (3)  k'  <  al 

79.  If  to  the  numerator  of  a  certain  simple  fraction  a  be  added, 

the  result  is  - ,  and  if  to  the  denominator  a'  be  added,  the 
result  is  — ;  what  is  the  original  fraction  ? 
Show  what  relations  must  exist  between  the  constants  so  that 


c    c 


-,  —  shall  be  simple  fractions  and  in  their  lowest  terms. 
d    cV 

Give  numerical  illustrations. 


350  EQUATIONS.  [XI. 

80.  In  a  certain  two-digit  number  the  second  digit  is  a  times 

the  first,  and  if  h  be  added  to  the  number,  the  digits  are 

reversed. 
Show  that  a  may  not  exceed  9,  nor  be  negative;  and  show 

when  a  may  be  fractional. 
Show  that  6  is  a  multiple  of  9  ;  and  show  what  bounds  h  lies 

between  for  different  values  of  a. 

81.  A  yacht  steams  up  a  river  m  miles  and  down  the  river  n 

miles,  in  h  hours  ;  again  she  steams  up  the  river  ??i'  miles 
and  down  the  river  n'  miles,  in  h'  hours  ;  what  is  the  rate 
of  the  yacht  in  still  water,  and  what  the  current  of  the 
river,  the  speed  of  the  yacht  and  the  current  of  the 
river  being  uniform? 
Give  numerical  illustrations  and  discuss  all  possible  cases. 

82.  A  dealer  has  three  kinds  of  tea,  worth  25  cents,  50  cents, 

and  one  dollar,  a  pound ;  how  shall  he  mix  them  by 
even  pounds  so  that  50  pounds  of  the  mixed  tea  shall 
be  worth  630? 

83.  Two  vases  a,  b  hold  v,  v'  gallons,  and  are  each  filled  with 

a  mixture  of  wine  and  water,  a  in  the  proportion  m :  ti, 
B  in  the  proportion  m' :  n'.  Two  other  vases  c,  d  are  of 
equal  size  and  hold  less  than  a  or  b  ;  c  is  filled  from  a, 
and  D  from  b  at  the  same  time  ;  c  is  emptied  into  b,  and 
D  into  A ;  and  then  the  proportion  of  wine  to  water  is  the 
same  in  a,  b  ;   of  what  size  are  the  vases  c,  d? 

84.  Of  two  ingots  the  first  has  a  parts  gold,  6  parts  silver,  the 

second  has  a'  parts  gold,  b'  parts  silver ;  in  what  propor- 
tion  shall  they  be  combined  so  that  the  product  shall 
have  c  parts  gold,  d  parts  silver? 
Between  what  bounds  do  c,  d  lie  ? 

85.  If  A  can  do  a  units  of  work  in  a'  days,  b,  b  units  in  6' 

days,  c,  c  units  in  c'  days ;  in  how  many  days  can  they 

do  a  +  6  -f  c  units,  all  working  together? 
What  is  the  value  of  c'  if  the  whole  work  be  finished  in  h 

days  ? 
Give  numerical  illustrations. 


§  19.]  EXAMPLES.  851 

86.  To  do  a  certain  work  a  needs  m  times  as  long  as  b  and  c, 

B  n  times  as  long  as  c  and  a,  c  p  times  as  long  as  a  and 
B  ;  find  the  relation  between  m,  ti,  p. 

87.  Two  right  triangles  have  their  right  angles  coincident,  and 

the  sides  about  that  angle  extend  along  a  horizontal  line 
a,  a'  yards,  and  along  a  vertical  line  &,  6'  yards ;   find 
how  far  the  point  of  meeting  of  the  two  hypothenuses 
lies  to  the  right  and  above  the  vertex  of  the  right  angle. 
Discuss  all  the  possible  cases. 

88.  The  points  a,  b,  c,  •••  lie  on  a  straight  line,  at  distances  a, 

6,  c  •••  from  a  fixed  point  o  upon  the  line  ;  find  a  point 
X  on  this  line  such  that  its  distance  from  any  point  m 
on  the  line  shall  be  the  average  of  the  distances  of  a,  b, 
c  •••  from  M. 
Show  that  the  result  is  independent  of  m. 

89.  A  reservoir  is  filled  by  pipes  a,  b  in  c  hours,  by  pipes  b,  c 

in  a  hours,  by  pipes  c,  A  in  6  hours ;  in  what  time  is  it 
filled  by  each  pipe  running  alone  ?  and  by  all  three  running 
together  ? 

Give  numerical  illustrations  and  discuss  all  possible  cases. 

Show  what  relations  must  exist  between  a,  6,  c  so  that  no 
water  flows  through  either  a,  or  b,  or  c,  or  any  two  of 
them ;  and  what  relations  must  exist  so  that  one  or  two 
of  the  pipes  shall  give  an  outflow. 

90.  A  reservoir  holding  7)i  gallons  is  filled  by  two  pipes,  a,  b, 

running  a,  h  gallons  an  hour,  and  emptied  by  two  pipes, 
c,  D,  running  c,  d  gallons  an  hour.  What  is  the  rela- 
tion between  ci,  6,  c,  d,  so  that,  with  all  the  pipes  run- 
ning, the  reservoir  shall  be  filled  in  Ti  hours?  that  it 
shall  be  emptied  in  k  hours  ? 

So  with  one  pipe  running  in  and  two  out  ?  or  two  in  and 
one  out? 

Make  a  general  formula  to  involve  all  the  pipes,  counting 
the  outflow  as  negative  inflow. 

Give  numerical  values  to  the  letters,  and  interpret  the 
results  in  special  cases. 


352  EQUATIONS.  [XI. 

91.  A  hound  pursues  a  fox,  aud  makes  a  leaps  while  the  fox 

makes  b  leaps  ;  but  c  hound-leaps  equal  d  fox-leaps. 
The  fox  has  a  start  of  7c  leaps  ;    when  will  the  hound  over- 

take^the  fox? 
Give  numerical  values  to   the  letters,  and  interpret   the 

results. 
What  is  the  relation  between  a,  6,  c,  d,  so  that  the  hound 

shall  never  catch  the  fox  ?  what  the  relation  so  that  the 

fox  is  running  away  from  the  hound?  what  the  relation 

so  that  the  hound  will  catch  the  fox  ? 

92.  Two  couriers,  a,  b,  are  at  m,  n,  d  miles  apart,  and  going 

forward  at  a,  b  miles  an  hour ;  when  are  they  together? 
Consider    the    following    cases :     (a)    when    they    move 

towards  each  other ;   (b)  when  away  from  each  other ; 

(c)  when  in  the  same  direction,  a  behind  b,  and  going 

(1)  faster  than  b,  (2)  slower,  (3)  at  the  same  rate. 
Interpret    the    several    results,    and    illustrate    by   giving 

numerical  values  to  a,  6,  d. 

93.  Three  couriers,  a,  b,  c,  are  all  upon  the  same  straight  road, 

and  going  at  a,  6,  c  miles  an  hour.  They  are  now  at  the 
points  M,  N,  p,  distant  m  to  n,  h  miles,  n  to  p,  A:  miles. 

Find  when  a  will  be  midway  between  b,  c ;  b  midway  be- 
tween c,  A  ;  c  midway  between  a,  b. 

Show  what  special  relations  must  exist  between  a,  6,  c,  so 
that  they  may  be  all  together. 

Take  distances  to  the  right,  and  time  forward,  positive ; 
distances  to  the  left,  and  time  past,  negative.  Find  the 
general  formulae  ;  and  interpret  the  several  results  when 
differeut  numerical  values  are  given  to  the  letters. 

94.  If  the  hour  and  minute  hands  of  a  clock  be  together  at 

12  noon,  at  what  other  times  between  noon  and  mid- 
night will  they  be  together?  at  what  times  will  they  be 
opposite  to  each  other?  at  what  times  will  they  be  at 
right  angles  to  each  other? 
Apply  the  fact  that  the  minute  hand  gains  55  minute  spaces 
in  GO  minutes. 


§  19.]  EXAMPLES.  353 

95.  If  the  hour,  minute,  and  second  bands  of  a  clock  all  turn 

on  the  same  pivot,  and  if  they  be  together  at  12  noon, 
when  will  they  next  be  together?  at  what  times  will  the 
second  hand  be  midway  between  the  other  two  ?  at  what 
times  will  the  minute  hand  be  midway  between  the  other 
two  ?  at  what  times  will  the  hour  hand  be  midwa}'  between 
the  other  two?  at  what  times  will  they  divide  the  clock 
face  into  three  equal  spaces  ?  at  what  times  will  they  form 
a  T  with  either  hand  as  the  body,  and  the  other  two  as 
the  head  of  the  T? 

96.  If  three  planets,  a,  b,  c,  circle  about  the  sun  in  the  same 

direction,  and  with  orbits  in  the  same  plane,  in  a,  6,  c 
years,  and  if  they  be  now  in  conjunction  (on  the  same 
side  of  the  sun,  and  all  in  a  straight  line  with  it),  when 
will  they  be  again  in  conjunction?  when  will  a,  b  be  in 
conjunction,  and  c  in  opposition?  when  will  b,  c  be  in 
conjunction,  and  a  in  opposition?  when  will  c,  a  be  in 
conjunction,  and  b  in  opposition?  when  will  they  so 
stand  that  the  arc  ab  subtends  an  angle  6  at  the  sun, 
the  arc  bc  an  angle  0\  the  arc  ca  an  angle  ^"  ? 

97.  Three  boys,  a,  b,  c,  starting  together,  run  round  a  circu- 

lar m-yard  track,  at  a,  6,  c  yards  a  second ;  find  general 
formulae  for  the  times  of :  their  conjunction,  the  con- 
junction of  two  of  them  and  opposition  of  the  third, 
the  division  of  the  track  into  arcs  d,  e,  /,  such  that 
d-^e  +/=  m  ;  first  when  all  run  in  the  same  direction, 
second  when  two  run  in  the  same  direction  and  one  in 
the  opposite  direction. 

Show  that  the  last  formulae  are  identical  with  the  first  if 
the  speed  of  the  one  be  called  negative. 

Show  what  relations  must  exist  between  a,  &,  c,  that  the 
runners  may  never  again  be  all  together. 

§   11,  PEOB.  7. 

•  ••101.    Find  the  values  of  x  from  the  equations  : 

98.  (x2+l)(a^+2)  =  (a^  +  6)(a^-l). 

99.  i(a^  -  la^-)  -  \{x'  -  \a?)  +  \(x^  -  j\a')  =  0. 


354  EQUATION?  S.  [XL 

100.  iiSx"-  7)  H-  J  (25  -  4a;2)  =  J (oa;^  -  14) . 

101.  2(2ar^-5)-i  +  (ar-3)-i=G(3ic2-l)-\ 

§    11.   PEOB.   8. 

•••  117.    Solve  both  by  completing  the  square  and  by  factoring : 

102.  ic2_8a;-f  15  =  0,     x^ -\- 10x  =  -24t,     a^-5a;4-4  =  0. 

103.  6a^-19a;+10  =  0,    7ar-3a;  =  160,     110ar^-21a;+l=0. 

104.  (5a;-3)--7=44a;H-5,    (3a;-5)  (2a;-5)  =  (a;+3)(a;-l). 

105.  ^a^+ix-\-^^^  =  0,    (a;-2)-i-2(a;  +  2)-i  =  |. 

1^^     3a;  — 2      2a;  — 5      8      a;  +  3  ,  a;  — 3      2a;  — 3 

10b. =-,     — ■ = • 

2x-6      3a;-2      3      a;  +  2      a;-2       a;-l 

107  ^  +  «  .  x  +  b_a      b      a;  +  a     a;  +  6      ^  +  ^=^3 
x—a     x—b     b      a     x—a     x—b     x—c 

108.  ar-(5+30a;H-i(ll  +  130=0,  ar^_(44-3i)a;+(7+5i)=0. 

109.  3a;4-2V-K-l=0,    ar*- 13 a?r«=:  14. 

110.  a;*-14a;2_^4o  =  0,    a;*  +  f  a;~*  =  3],    V2aJ-7a;  =  - 52. 

111.  a;  +  5— V(^  +  5)  =  6,    ^x-\-^x-^  =  2\. 

112.  V(2aJ+7)  +  V(3aJ-18)  =  V(7a;+l). 

113.  a;  +  V^  +  V(^  +  2)  +  V(a^+2a;)  =  a. 

114.  a;2+3=2V(^-2a;+2)+2a;,  ^{a?-2x-{-'d)-\x'=Z-x. 

115.  3ar+15.'c-2V(a^  +  5a;+l)  =  2. 

116.  a;2-2V(3a;'-2aa;  +  4)  =  -|a(a;  +  ia  +  l). 

117.  nQ?  +  x-\-n  +  l  =  0,    x^ -\-a?  —  Ax  — 4:  =  Q. 

118.  Form  the  quadratic  equations  whose  pairs  of  roots  are  : 

2,3;    1,-4;    3±2i;    -l±i;    ±3+2i;    4+5^,   1  +  21. 

119.  Form  equations  by  putting  equal  0  the  quadratic  functions  : 

2o?-\-x-(j;    6a;2_j9^^i5.    r^^_2mx  +  iiv-n^ \ 
a?—  (m  +  7i)  a;  +  (m  -\-p)  {n  —p)  ;     (a;  —  o)^—  6^ ; 
a;2—  5(1  +  i)x  +  13 1,    ar^  +  (7  +  50^;  4-  6  +17i. 
Solve,  and  b}'  aid  of  the  roots  factor  the  functions. 

120.  If  a  and  jB  be  the  roots  of  the  equation     x^—px  +  g  =  0, 

find  the  value  of  ap-^  -\-  (Ba-^  and  of  a?  +  ^^ 

121 .  What  value  of  c  gives  the  equation   5a;^  +  3a;  +  c  =  0 

equal  roots  ? 


§  19.]  EXAMPLES.  355 

122.  Eliminate  x  from  the  two  equations  * 

aQcF-\-bx-^c  =  0,     a'x-+6'aj  +  c'=  0. 

123.  Show  that  the  quadratic  function   ax^-j-bx-{-c  may  be  writ- 

ten in  the  forms   —  {  (2  ax  +  bY—  (5^—  4  ac)  ] 

and  -^{2ax+b  +  ^{b--4ac)  l\2ax  -\- b  - -^  (b'- -  A  ac)  } ; 

hence  derive  the  condition  for  real  and  unequal,  for  equal, 
and  for  imaginary  factors.  By  this  method  factor  the 
function  3 a^H-  5ic  +  2,  and  find  for  what  values  of  x  the 
function  vanishes. 

§  12,  PROB.  9. 

124.  Plat  the  quadratic  functions  ;  hence  find  the  real  values  of 

X,  if  any,  that  make  these  functions  vanish : 

oy^^Ax  +  S,  a^  — 4aj  +  3i,  ic^  — 4a;  +  4,  a.-^  — 4aj  +  4i, 

x-  +  x  +  (j,   -X--X  —  G,   -3ar^-10a;+13. 

§   13,    PROB.  10. 

125.  Find  five  couvergents  to  the  roots  of  the  equations  : 

x^-\-x-6=0,  x^-Sx  +  2  =  0,  a;2-Ga;  +  9  =  0, 
3a;-  +  4«=7,  4ar  — 3a;=10,    oa;-— 10a;=  20. 

§   14,    PROB.  11. 

126.  Find  the  maximum  or  minimum  values  of  the  functions 

ar^_4a;  +  3,  10  +  4a;-a^,  a^-6x  +  9,   -aj'^  +  6a;-9; 
and  the  correspondnig  values  of  x. 

127.  From  the  plat  of  the  functions  in  the  examples  of  §  12, 

state  which  of  them  have  maximum  values  and  which 
minimum,  and  find  these  values. 

Show  that  each  of  these  functions  has  a  -^      .\  value  if 

'  mmimum 

the  vertex  of  the  corresponding  parabola  be  -{  ^^^^        \ 

i.e.,  if  the  coefficient  of  x-  be^  positive. 
'  '  negative. 

128.  Show  that  - — ^-—  has  no  value  between  1  and  5. 

a;  — 3 

129.  Find  the  maximum  value  of  (x  +  a)  (x  —  b)  :  a^. 

130.  Show  that  a(a-\-x)  :  (a  —  x)  can  have  any  value. 


356  EQUATIONS.  [XI 

131.  Find  the  maximum  or  minimum  values  of 

ar-^  2x-\- 11 .     ar  —  x-\-l  ,     a-\-x  ,  a  —  x 
ar  4- 4ic-|- 10  '     ar-j-a;  — l'     a  —  x     a-^x' 
To  the  last  apply  the  principle  that  if  the  product  of  two 
variables  be  constant,  their  sum  is  a  minimum  when  they 
are  equal. 

132.  Prove  that  the  quotient  (x -{- a) :  (x"^ -^  bx -\-  c^)  always  lies 

between  two  fixed  finite  bounds  if  o?  +  cr>ab  and  6^<4c- ; 
that  tliere  are  two  bounds  between  which  it  cannot  lie  if 
a^-\-€^>ab  and  b^>4:Cr;  and  that  it  may  take  all  values 
if  a--\-(^<ab. 

133.  Find  what  value  of  x  will  make  a  maximum  the  product : 

(ar^  +  2a;+l).(7~ar^-2a;)  ;    (a^^  -  25)  :  (25  -  a^) . 
Apply  the  principle  that  if  the  sum  of  two  variables  be  con- 
stant, their  product  is  a  maximum  when  they  are  equal. 

134.  Find  the  sides  of   the  maximum  rectangle  that  can  be 

inscribed  in  a  given  circle. 

§  15,  PBOB.  12. 

•  ••  168.    Find  the  values  of  x,  y  from  the  pairs  of  equations  : 

135.  x-\-y=l,    a^-f2r  =  34;     x-y=l2,    x-  +  y''  =  74:. 

136.  x-\-y  =  a,    xy  =  b'^\     x  —  y  =  a,    xy  =  b". 

137.  Zx  —  by=2,    ary=l;     a;  +  ?/  =  100,    a;?/ =2400. 

138.  x-\-y  =  a,   x^-\-f  =  W\    a;^  +  ?/~2  =  4,   x-y-^=^. 

139.  a; +  2/ =  4,   x-^+y-^=l',     2x  +  ^y  =  d>l,   x-^-\-y-^=\^. 

140.  x  +  y  =  2,  ix?—2xy  —  y-  =  l',    x  +  y  =  \%,  a? ^f^AQll. 

141.  a;  +  2/=72,  ^a;  +  ^2/=6;   x'y-' + fx-' =  ^ ,  x-^+y-^=.%. 

§  15,  PBOB.  13. 

142.  4a;2_|_  72,2  ^143^    3a^-/=ll;   x-\-y  =  a?,   ^y-x=y\ 

143.  a^+/=fa!2/,   x  —  y  =  \xy',     x^-\-xy  =  Q,   a^^y  =  5. 

NOTE    1. 

144.  ar  + a;?/ +  2?/- =74,    2x^ +  2xy -^-y- =  73. 

145.  a:F-{-y^  =  a^,    xy  =  b-;      a^  — 2/^  =  a^    xy  =  b^. 

146.  x^-\-Sxy  =  o4:,    xy -{- Ay^  =  115  ;     a^-^f  =  9,   xy  =  2. 

147.  x'+xy-{-4:y^=6,  3xF  +  8f-=U',    x'-{-f-=l,  a^+2/«=l. 

XOTE   2. 

148.  x^-3xy-{-2y^=0,x^+y-=x'^-f;   a^-y^=8,x^-f=26. 

149.  x^-^xy-6y^=0,    2x-{-Sy +  x^-}-3xy ^2y^=19. 

150.  8a;3  4-3ar^2/-fa;/=18,  2a^+5aj22/+3a;/  =  24. 


§  19.]  EXAMPLES.  357 

XOTE    3. 

151.  a;+?/  =  5,    a^  +  2/^  =  97;    a;  — 2/  =  3,    af  — 2/^  =  3093. 

152.  x^  +  y^=:Uocry'-,    x-{-y=:9;     x^  +  y^  =  l,    a^  +  y  =  17. 

153.  x^  -\-y^  =  7  -{-  xy,    x^-\-y^=  Qxy -—1, 

NOTE   5. 

154.  a*  — a;2^2/^_2/2  =  84,    a;2^  3j22^2_^^2^  49^ 

155.  xy{x-\-y)=SO,    a^  + 7/^  =  35. 

NOTE    G. 

156.  4.{x-{-y)  =  Sxy,    x  +  y +  x^ -i-y^  =  26. 

157.  a^(a;  +  ?/)  =  80,    aj2^2a;- 3?/)  =  80. 

158.  a;^  +  a.'2?/2  + 7/4  =  133,    a;^  -  a;?/ -f  ^/^  =  7. 

159.  »2  +  /  — (a;  +  2/)=a,    x* -\-i/-]-x-\-y —  2{a^  +  f)  =  b. 
IGO.  a;4-2/-fV^y=14,    a;^  +  ?/- +  a-?/ =  84. 

161.  x^-{-y  =  4:X,  y--{-x  =  4:y;    af-{-xy^=10,  f  +  x^y  =  5. 

162.  a;3_^2r  +  3aj  +  32/  =  378,  a^  +  2/'- 3a;- 3.?/  =  324  ; 

ar=  ax'  +  6?/,  ?/-=  ay-}-bx;   bx  +  ay  =  ab,  bx-{-ay  =  Axy. 

163.  10a^+15a;?/  =  3a6-2a^  107/2  + 15a;?/ =  3a6- 2&^ 

164.  6a;2-3a;-42/=25,  a;2_^2a;-3?/=18; 

165.  xy-i-6x-{-7y==50,  Sxy +  2x+5y=72. 

166.  a; +  2/ =10,  V^^r' +  V2/^~'  =  l ; 

167.  V(^'  +  2/')+V(^'-/)=22/,    a;*-2/4  =  a^ 

168.  8a;*-?/-^  =  14,  a;t2/'  =  22/2. 

•  ••172.   Find  the  values  of  a;,  ?/,  z  from  the  sets  of  equations  : 

169.  yz  =  bc,  bx-\-ay  =  ab,    cx  +  az  —  ac. 

170.  a; +  2/ +  2;  =  37-^  +  2/"^ +  0-1  =  J,    x?jz=:l', 

171.  a;2/  =  a(a;  +  2/)5  a;2;  =  6 (a;  +  5;) ,  yz=:c(y  +  z). 

172.  a;  +  2/  +  2;  =  6,  4a;  +  ?/  =  22;,  x^ +  y' -\-z'=U. 

§10. 

173.  A  boat-crew  rows  3i  miles  down  a  river  and  back  again 

in  an  hour  and  40  minutes  ;  if  the  river  have  a  current  of 
two  miles  an  hour,  at  what  rate  does  the  crew  row  ? 

174.  A  number  is  composed  of  two  digits  ;  the  first  exceeds  the 

second  by  unity,  but  the  number  itself  falls  short  of  the 
sum  of  the  squares  of  its  digits  by  26  ;  what  is  the  number? 

175.  A  number  is  composed  of  two  digits;  the  first  exceeds 

the  second  by  2  ;  the  sum  of  the  squares  of  the  given 
number  and  of  the  number  got  by  reversing  the  digits  is 
4034  ;  what  is  the  number  ? 


858  EQUATIONS.  [XL 

17G.  Find  the  lengths  of  the  segments  of  a  line  a,  if  m  times 
the  square  of  one  be  equal  to  n  times  the  rectangle  under 
the  whole  line  and  the  other. 

177.  The  drivinsr-wheels  of  a  locomotive  are  2  feet  lonojer  in 

diameter  than  the  running-wheels ;  the  running-wheels 
make  140  turns  more  than  the  driving-wheels  in  a  mile  ; 
what  are  the  diameters?        [ratio  circum. :  diam.=  22  : 7 

178.  A  set  off  from  London  to  York,  and  b  at  the  same  time 

from  York  to  London,  and  they  traveled  uniformly ;  a 
reached  York  4  hours,  and  b  reached  London  9  hours, 
after  they  met ;  in  what  time  did  each  make  the  journey? 

179.  A  broker  bought  a  number  of  hundred-dollar  railway  shares 

at  a  certain  rate  discount  for  S  7500,  and  afterwards,  at 
the  same  rate  premium,  he  sold  them  all  but  GO  for  85000  ; 
how  many  did  he  buy,  and  what  did  he  give  a  share? 

180.  Divide  a  line  3  feet  long  into  two  parts  such  that  the 

circle  standing  on  one  segment  as  diameter  shall  be  equal 
to  the  square  standing  on  the  other. 

181.  The  number  5G3  in  the  decimal  scale  is  less   than   the 

same  number  in  a  higher  scale  by  232  ;  what  is  the 
radix  of  the  higher  scale  ? 

182.  What  is  the  price  of  eggs  when  two  more  in  a  shilling's 

worth  lowers  the  price  one  penny  a  dozen  ? 

183.  There  are  two  numbers  whose  product  is  the  difference  of 

their  squares,  and  the  sum  of  whose  squares  is  the  differ- 
ence of  their  cubes  ;  what  are  the  numbers  ? 

184.  The  sum  of  the  squares  of  the  numerator  and  denominator 

of  a  fraction  is  389,  and  the  difference  of  the  fraction 
and  its  reciprocal  is  \^^ ;  find  the  fraction. 

185.  Find  two  numbers  such  that  their  sum,  their  product,  and 

the  sum  'of  their  squares  shall  be  equal  to  each  other. 
18G.    Find  two  numbers  whose  product  is  p^  and  the  difference 

of  whose  cubes  is  m  times  the  cube  of  their  difference. 
187.    Find   a   fraction  the  product   of   whose   numerator    and 

denominator  is  180,  and  such  that  if  its  numerator  and 

denominator  be  each  increased  by  10,  its  value  is  doubled. 


§  19.]  EXAMPLES.  859 

188.  A  rectangular  space,  whose  length  and  breadth  are  42 

and  78  feet,  is  surrounded  by  a  ditch  5  feet  deep,  and 
capable  of  holding  220  tons  of  water  ;  what  is  the  breadth 
of  the  ditch,  counting  6  tons  of  water  for  a  cubic  fathom? 

189.  There  is  a  fraction  such  that  if  the  numerator  be  increased 

and  the  denominator  diminished  by  2,  the  reciprocal  of 
the  fraction  is  the  result ;  but  if  the  numerator  be  dimin- 
ished and  the  denominator  increased  by  2,  the  result  is 
less  than  the  reciprocal  by  1^^ ;  what  is  the  fraction? 
Solve  the  same  problem  in  general  terms,  replacing  2  and 

190.  Two  boys  set  off  from  the  right  angle  of  a  right-triangular 

field,  running  in  opposite  directions,  with  speeds  in  the 
ratio  of  13  :  11  ;  they  first  meet  at  the  middle  point  of  the 
hj'pothenuse,  and  again  at  a  point  30  yards  distant  from 
the  starting-point ;  find  the  lengths  of  the  three  sides. 

191.  Two  cubical  vessels  together  hold  407  cubic  inches  ;  when 

one  vessel  is  placed  on  the  other,  the  total  height  is  11 
inches  ;  find  the  contents  of  each. 

192.  A  number  consists  of  two  digits,  the  difference  of  whose 

squares  is  40,  and  if  it  be  multiplied  by  the  number  con- 
sisting of  the  same  digits  taken  in  reverse  order,  the 
product  is  2701  ;  find  the  number. 

193.  A  vessel  can  be  filled  with  water  by  two  pipes;  by  one 

of  these  pipes  alone  the  vessel  would  be  filled  2  hours 
sooner  than  by  the  other ;  and  the  vessel  can  be  filled  by 
both  pipes  together  in  1 J  hours  ;  find  the  time  that  each 
pipe  alone  would  take  to  fill  the  vessel. 

194.  A  vessel  is  to  be  filled  with  water  by  two  pipes ;   the 

first  pipe  is  kept  open  during  three-fifths  of  the  time 
which  the  second  would  take  to  fill  the  vessel ;  then  the 
first  pipe  is  closed  and  the  second  is  opened ;  had  both 
pipes  been  kept  open,  the  vessel  would  have  been  filled  6 
hours  sooner,  and  the  first  pipe  would  have  brought  in 
two-thirds  of  the  water  which  the  second  pipe  did  bring ; 
how  long  would  each  pipe  alone  take  to  fill  the  vessel  ? 


300  EQUATIONS.  [XI. 

195.  A  number  consists  of  three   digits ;   the  first  is  to   the 

second  as  the  second  is  to  the  third ;  the  number  itself 
is  to  the  sum  of  its  digits  as  124  to  7;  and  if  594  be 
added  to  it, the  digits  are  reversed ;  what  is  the  number? 

196.  The  diagonal  of  a  box  is  125  inches,  the  area  of  the  lid  is 

4500  square  inches,  and  the  sum  of  three  conterminous 
edges  is  215  inches  ;  find  the  lengths  of  these  edges. 

197.  One  side  of  a  room  is  5  feet  longer  than  the  other  side, 

and  1000  square  feet  of  paper  is  needed  to  cover  its 
walls  ;  if  it  were  3  feet  higher,  the  same  paper  would  be 
needed  for  3  only  of  its  walls,  the  bare  wall  being  one  of 
its  longer  sides  ;  what  are  the  dimensions  of  the  room  ? 

§   17.     PROB.  14. 

198.  Solve  the  binominal  equations  :     ic^  —  1  =»  0,   a;^  -f  1  =  0, 

3^  =  -S,    a^  =  lG,    x^«+l=0,    a;^2^1  =  0,    ic^-l=0. 

199.  Find  the  square  root  to  three  decimal  places  of : 

5  +  12i,    12+5i,    161-240t,     13  +  7  i,     7  +  13^. 

200.  Prove  that  the  n  roots  of  the  equation  a"*  =  a  +  bi,  are 

all  given  by  the  expression 


yr  ••[  cos  —^ h  I  sm  — i— 

\  n  n 


wherein  r  is  the  tensor  and  6  the  versorial  angle  of  the 
number  a-f-6t',  and  k  has  any  ?i  consecutive  values  in 
the  series  of  natural  numbers  between  ~oo  and  +oo  . 

§  18. 

201.  Find  the  value  of  x  from  the  exponential  equations  : 

x+2  x+1 

2''=8,  2'+«  =  8^-3,  3i^2  =  27i^i,  9' =  3,  8^^1  =  2. 

202.  By  aid  of  the  table  of  logarithms  find  x  from  the  equations  : 

10^  =  3,  4^  =  10,  .3==  =.8,  32^+3^100*  ^  15^^+^  =  27^^-^ 

203.  Solve  the  equations  :    3^^-7.3^=18,  2*^*- 5.2^^  + 6  ==  0, 

2^+1  +  4'=  =  80,    4.32'+i- 5.3^+2  ^12. 

204.  If  ak'^^"^  4-  2  6Z:"=+~  +  c  =  0,    prove  that 

X  =  [logj  -  &fc"  ±  ■^ih-k''^  -  ack"^)  I  -log(aA;"')]  :  r  log k. 


Xll.th.  1,  §1.]  ARITHMETIC   PROGRESSION.  3G1 


XII.     SERIES. 

For  definition  of  series,  see  I.  §  12.     The  first  and  last 
terms  of  a  series  are  its  extremes;  the  other  terms  are  its  means. 

§  1.     ARITHMETIC    PROGRESSION. 

An  Arithmetic  Progression  is  a  series  such  that  each  term 
after  the  first  is  formed  by  adding  a  constant  to  the  next  pre- 
ceding term.     The  constant  added  is  the  common  difference. 

The  abbreviations  are  :  a  for  first  term,  I  for  last  term,  d  for 
common  difference,  n  for  number  of  terms,  s  for  sum  of  all 
the  terms. 

E.g.,  1,  3,  5,  7,  9,  is  an  ascending  series, 

wherein  cZ  =  ''"2,    a=l,    Z  =  9,    n  =  5,    s=25. 

So,  9,  7,  5,  3,  1,  ~1,  ~3,  is  a  descending  progression, 

wherein  d  =  -2,    a  =9,    Z=-3,    n=7,    s=21. 

Theor.  1.     In  an  arithmetic  progression 

1]  l  =  a-\-{n-l)d. 

For     •.•  a-\-d,    a  +  2d,    a  +  3c?,  "•,  a-{-{k  —  l)d 

are  the  2d,  3d,  4th,  •••  A:th  terms,  [df. 

.*.  a  +  (w  — l)d  =  Z,  the  last  of  a  series  of  n  terms,     q.e.d. 

Cor.     In  an  arithmetic  progression 
2]  a=l-{n-l)d, 

3]  d  =  l^, 

n—  1 

4]  w  =  — ; f-1. 

d 

The  reader  may  prove,  solving  formula  1  in  turn  for  a,  c?,  n. 


362  SERIES.  .  [Xll.th.  2, 

Theor.  2.     In  an  arithmetic  progression 
5]  s  =  \n{a  +  l). 

For       •.•  s  =  a-h(a4-d)4-(a+2d)H \-{l  —  d)-\-l,    n  terms, 

and  s  =  Z  +  (Z  — d)  +  (Z  — 2d)H \-{a-\-d)-\-a^  n  terms, 

.-.  2s  =  (a  +  Z)4-(a  +  Z)  +  (^  +  0H l-(a+0'  n  times, 

=  71  •  (a  -h  Z) . 
s  =  \n{a-\-l).  Q.E.D. 

Cor.  1 .     In  an  arithmetic  progression 

6]  a  =  ^-', 

7]  «=^-a, 

8]  n  =  ^. 

The  reader  may  prove,  solving  formula  5  in  turn  for  a,  L  n. 

Cor.  2.     In  an  anthmetic  progression 
9-1  ^  ^2s4-y^(n  — l)d 

2n 

10]        d  =  2M=:fl, 

n(w  — 1) 

111  ^^_cZ+2Z±vr(2Z  +  c^)^-8<fe] 

^  2d  ' 

12]  s=iw[2Z-(w-l)d], 

13]  ^  ^  2s  -  71(71 -1)(^ 

■^  271 

14]  ^^2(.-an) 

-^  71(71-1) 

15-]  ^^d-2a±vr(2a-cZ)^  +  8^g]^ 

16]  s=i7i[2a  +  (n-l)cZ], 

17]  a==ild±^l{2l-\-dy-8ds']\, 

18]  Z  =iJ-cZ±V[(2a-cZ)'  +  8(Zs]^, 

ion  ,  _  (Z  4-  g)  (Z  -  ct) 

^^J  ^-    2.-(Z  +  a)  ' 


20] 


{l  +  a)(l  —  a-hd) 
2d 


pr.  1,  §  1.]  ARITHMETIC   PROGRESSION.  3f>^ 

The  reader  may  prove  formulae  9-12,  combining  1,  5  so  as  to 
eliminate  a,  then  solving  in  turn  for  ?,  d,  n,  s;  formulae  13-16, 
by  eliminating  Z,  then  solving  for  a,  d,  n,  s;  formulae  17-20, 
by  eliminating  ?i,  then  solving  for  a,  I,  (?,  s. 

Note  1.  The  formulae  involving  a  may  be  got  from  those 
involving  Z,  and  vice  versa,  by  symmetry,  writing  a  in  place  of  Z, 
I  in  place  of  a,  and  —  d  in  place  of  -{-d;  and  thus  seven  of  the 
fourteen  formulae  1,  2,  6,  7,  9-18  may  be  written  directly  from 
the  other  seven  ;  for  if  any  arithmetic  progression  be  reversed, 
then  a  becomes  I,  I  becomes  a,  and  d  becomes  —  d. 

Note  2.  Formulae  11,  15  give  two  values  for  n.  If  either 
of  these  values  be  negative  or  fractional,  it  may  be  rejected  as 
inconsistent  with  the  conditions  of  the  problem.    [XI.  pr.  6  nt.  3 

PrOB.  1.      To   INSERT  m  ARITHMETIC    MEANS   BETWEEN  «,  I. 

Divide  the  remainder,  1  —  a,  6?/  m  4- 1  for  the  common  differ- 
ence; and  to  a  add  one,  two,  three,  •••  times  this  difference. 

E.g.,  to  insert  5  means  between  12  and  48  : 
then    •.•   (48  —  12)  :  (5  +  1)  =  6,  the  common  difference, 
.-.  the  series  sought  is  12,  18,  24,  30,  36,  42,  48. 

Note.  By  aid  of  this  problem,  from  every  arithmetic  pro- 
gression a  new  arithmetic  progression  may  be  formed  by  insert- 
ing the  same  number  of  arithmetic  means  between  ever^-  two 
consecutive  terms  ;  and  the  common  difference  of  this  new  pro- 
gression is  the  quotient  of  the  common  difference  of  the  other 
divided  by  one  more  than  the  number  of  terms  so  inserted. 

So  from  any  arithmetic  progression  a  new  progression  may 
be  formed  by  taking  equidistant  terms  ; 

E.g.,  if  two  means  be  inserted  between  two  consecutive  terms  : 
then         6,  12,  18,  24,  30,... 

becomes  6,  8,  10,  12,  14,  16,  18,  20,  22,  24,  26,  28,  30,  .•• 
and  if  of  this  new  progression  the  first,  fifth,  ninth,  .••  terms 

be  taken,  a  third  progression  is  formed, 
6,     14,     22,     30,  ... 
whose  common  difference  is  4 . 2,  =  8. 


364  SERIES.  [XII.  ths. 

§2.     GEOMETRIC   PROGRESSION. 

A  Geometric  Progression  is  a  series  such  that  each  term 
after  the  first  is  formed  bj^  multiplying  the  next  preceding  term 
by  a  constant  multiplier.     The  multiplier  is  the  common  ratio. 

The  abbreviations  are  :  a  for  first  term,  I  for  last  term,  r  for  com- 
mon ratio,  n  for  number  of  terms,  s  for  sum  of  all  the  terms. 

Ti-,  I  >  1     4.U         •      .     I  an  ascendinq 

When  r  ^  ^  1,  the  series  is  ^  ^  ^^^^^,^^.,^^  progression. 

E.g.^  1,  2,  4,  8,  16,  is  an  ascending  series, 

wherein  r=''"2,    rt=l,      ?  =  16,    7i  =  5,    s=31. 

So  1,  ~2,  4,  ~8,  16,  is  an  ascending  series, 

wherein  7-=~2,    a=l,      ?=16,    ?i  =  5,    s=ll. 

But  16,  8,  4,  2,  1,  1^,  :^,  is  a  descending  series, 

wherein  r=^,      a  =16,    Z  =  J,      n=7,    s  =  31f. 

Theor.  3.     In  a  geometric  progression 

21]  l  =  a7^-^. 

For    • .  •  ar^  ar^^  ar^,  •  •  •  ar*~^  are  the  2d,  3d,  4th,  •  •  •  A;th  terms,  [df . 
.*.  ar^~^  =  l,  the  last  of  a  series  of  n  terms.       q.e.d. 

Cor.  In  a  geometric  progression 

22]  a  =  l:r^-\ 

23]  r==*-^{l:a), 

«JT  1   .  loffZ  — loga 

24]  n  =  1  H — ^- ^-  • 

logr 

The  reader  may  prove,  solving  formula  21  in  turn  for  a,  r,  w. 
Theor.  4.     In  a  geometric  progression 


4-  ar^'^  4-  ar"? 


Q.E.D. 

r  — 1  1  — r 


25] 

s 

ar^  —  a 

-a^-*" 

r-l 

1— r 

For 

•.• 

8 

=  a  -\-ar 

+  ar'+>> 

.*. 

rs 

=  ar  +  ai^-har^-^-' 

,♦. 

rs 

—  s  =  af^ 

-a, 

, 

s 

ar^  —  a 

-a'-^. 

3, 4,  §  2.]  GEOMETRIC  PEOGRESSION.  365 

Cor.  1 .     In  a  geometric  progression 
26]  a  =  fc=ii^, 

27]  a?'*  —  sr  =  a  —  s^ 

28]  n  =  logO'g-^  +  a)-loga 

logr 

The  reader  may  prove,  solving  formula  25  in  turn  for  a,  r,  n. 
He  will  observe  that  formula  27  [r  unknown]  is  of  the  Tith 
degree,  for  which  there  is  no  general  solution.  In  numerical 
equations  the  solution  is  always  possible. 

Cor.  2.  In  an  infinite  decreasing  geometric  progression^  the 
limit  ofv^  is  0  ;  and  the  value  of  8  is  the  quotient  a  :  (1  —  r). 

CoR.  3.     In  a  geometric  progression 

29]  I  ={r-l)s.r--^^ 

30]  r» !_r"-i  +  -L.  =  o, 

s  —  I  s  —  I 

31]  ^^log^-logpr-(r-l)3]_^^^ 

logr 

32]  8=  ^^"^^, 

33]  a(s-ay-^  =  l{s-iy-\ 

34]  I  (s  -  ly-"-  =a{s-  ay-\ 

J  log(s-a)-log(s-0  ^    ' 

37]  a  =  lr-s(r-l), 

s(r—  l)  +  a 
r 
a 


36] 

37] 

38]  Z 


39]  r  = 

40]  s  = 


Ir  —  a 
r-i' 


366  SERIES.  [XII.  th.  6, 

The  reader  may  prove  formulae  29-32,  combining  formulae 
21,  25  so  as  to  eliminate  a,  tlien  solving  in  turn  for  l^  r,n,s; 
formulae  33-36  by  eliminating  r,  then  solving  for  a,  I,  n,  s ; 
formulae  37-40  b}'  eliminating  w,  then  solving  for  a,  I,  r,  s. 
He  will  observe  that  formuhe  30,  33,  34  have  no  general  solu- 
tions.    In  numerical  equations  their  solution  is  always  possible. 

Note.  The  formulae  involving  a  may  be  got  from  those 
involving  I,  and  vice  versa,  by  symmetry,  writing  a  in  place  of  I, 
I  in  place  of  a,  and  r~^  in  place  of  r"*"^ ;  and  thus  seven  of  the 
fourteen  formulae  21,  22,  25-34,  37,  38  may  be  written  directly 
from  the  other  seven ;  for  if  any  geometric  progression  be  re- 
versed, then  a  becomes  I,  I  becomes  a,  and  r"'"^  becomes  r~^. 

PrOB.   2?     To  INSERT  m  GEOMETRIC  MEANS  BETWEEN  a,  I. 

Take  the  (m  -f  l)th  root  of  the  quotient  1 :  a  foi'  the  common 
ratio;  and  multiply  a  by  the  first,  second -" powers  of  this  ratio. 

E.g.,  To  insert  three  means  between  3  and  48 : 
then   *.•   -v/(48  :  3)  =  2,  the  common  ratio, 
.-.  the  series  sought  is  3,  6,  12,  24,  48. 

Note.  By  aid  of  this  problem,  from  every  geometric  progres- 
sion a  new  geometric  progression  may  be  formed  by  inserting 
the  same  number  of  geometric  means  between  every  two  con- 
secutive terms ;  and  the  common  ratio  of  this  new  progression 
is  that  root  of  the  common  ratio  of  the  other  whose  index  is 
one  more  than  the  number  of  means  so  inserted. 

So,  from  any  geometric  progression  a  new  progression  may 
be  formed  by  taking  equidistant  terms. 

E.g.,  if  two  means  be  inserted  between  two  consecutive  terms, 
then         3,  6,  12,  24,  ••• 

becomes  3,  3^2,  3^4,  6,  6^2,  6^4,  12,  12-^2,  12^4,  24,.-., 
and  if  of  this  new  progression  the  first,   fifth,  ninth,  ••• 

terms  be  taken  a  third  progression  is  formed 
3,  6^2,  12^4,  - 

whose  common  ratio  is  the  fourth  power  of  either  of  the  three 
values  of  -^'2. 


prs.2,3,  §3.]  HARMONIC   PKOGRESSIOX.  367 

§  3.     HAKMONIC   PROGRESSION. 

A  Harmonic  Progression  is  a  series  such  that  any  three 
consecutive  terms  being  taken,  the  ratio  of  the  first  to  the  third 
equals  the  ratio  of  the  excess  of  the  first  over  the  second  to  the 
excess  of  the  second  over  the  third. 

E.g.,,  if  p,  q,  r,  be  any  three  consecutive  terms  of  a  harmonic 
progression,    then   p:  r=p  —  q  :  q  —  r. 

Theor.  5.  If  a  series  of  numbers  be  in  harmonic  progression 
their  reciprocals  are  in  arithmetic  progression;  and  conversely. 

Let  p,  q,  r  be  any  three  consecutive  terms  of  a  harmonic 
progression ; 
then  will  r~^  —  q~^  =  q~^  —  i>~^« 

For     '.'  p:r=p  —  q:  q —  r,  [df. 

.'.pq—pr—pr  —  qr,  [II.  th.  6 


.  r 


■1      ^-1 ^-1 


q~^  =  q~^—p~^i  Q.E.D.     [div.bypgr 

So  for  the  converse. 

PrOB.  3.      To   INSERT   m   HARMONIC   MEANS   BETWEEN   TWO   EX- 
TREMES, a,   I. 

Find  m  arithmetic  means  between  a~^  and  1~-^,  and  take  their 
reciprocals. 

E.g.,  to  insert  two  harmonic  means  between  12  and  48  ; 
then   •.•   j\-^\=:^-g,    and   ^\:S=^\, 

.*.  the  arithmetic  progression  is    yV?  iV?  yV'  ts'       [P^-1 

and  the  harmonic  progression  is     12,  16,  24,  48,       [th.  5 

wherein    12:24=12-16:16-24,    16:48=16-24:24-48. 

Note.    The  analogies  and  relations  of  the  three  progressions 

appear  below  :   If  p,  q,  r  be  three  numbers 

I  arithmetic  Ip'-P't 

in  <  geometric  progression,   then  p  —  q\q  —  r—  \p'.q\ 
I  harmonic  |  p  :  r ; 

I  arithmetic  I  \{p-\-r'). 

and  the  \  geometric  mean  of  p,  r  \s,  \  -yjpr. 

I  harmonic  |  2pr :  (p  +  r) . 

So,  the  geometric  mean  of  jvf"is  the  geometric  mean  of  the 
arithmetic  and  harmonic  means  of  p,  r. 


368  SERIES.  [XII.  ths. 

Theor.  G.  If  four  numbers,  p,  q,  r,  s,  be  so  related  that 
p  —  q,  p  —  r,  p  —  s  form  a  harmojiic  progression,  then : 

(a)  q  —  r,  q  —  s,  q—  p  likewise  form  a  harmonic  progression ; 
and  so  do  r  —  s,  r  —  p,  r  —  q  ;  and  s  —  p,  s  —  q,  s  —  r. 

(6)  Tlie  relations  between  p,  q,  r,  s  shown  in  (a)  hold  true  also: 

1.  Among  any  four  numbers,  n+p,  n+q,  n+r,  n-f-s,  ichose 
differences  equal  the  differences  of  p,  q,  r,  s  ; 

2.  Among  any  equimultiples  of  p,  q,  r,  s  ;  or  o/  their  reciprocals; 

3.  Among    ^-^.   ^.    ?^,    ^, 

cp  -f  d     cq  -f-  d     cr  H-  d     cs  -f-  d 

wherein  a,  b,  c,  d  are  any  numbers. 

(a)  '.'  the  condition  that  — ^^ 1 — =  -^  [th.  5 

%  p—qp—s      p—r 

is  that      (p  +  r)  •  {q-\-s)  =  2pr  -\-2qs;       [free  fr.  fracts.,  red. 

112 
and    •.*  the  condition  that h 


q  —  r       q—p       q  —  s 
is  that       {q  +  s)'{r+p)  =  2qs  -{-2rp,     Ich.  p,q,r,s  to  q,r,s,p 
I.e.,  that  {p-\-r)'{q-\-s)  =  2pr-\-2qs,    as  above, 

.-.  when  p—q,  p—r,  p—s  form  a  harmonic  progression, 
so  do  q  —  r,  q  — s,  q—p.  q.e.d. 

Sodor  — s,  r—p,  r  —  q;    and    s—p,  s  —  q,   s  —  r. 
(b)  •••  relation  (a)  involves  p,  q,  r,  s  only  by  their  differences, 
.-.  it  holds  for  any  numbers  n-\-p,n-\-q,n-j-r,n-\-s.q.i:.'D. 

2.  • .  •  equation  (p  +  r)  •  (g  +  s)  =  2pr  +  2qs    is  not  changed 

when  for  p,  •••  s  are  put  np,  •••  ns;  or  n:p"-  n:  s', 
.'.  the  equation  is  true  for  these,  if  for  J),- ••  s.  q.e.d.  [above 

3.  ...  cip±b^a_^bc-ad^    m±l  =  ...^    ...,       [division 

cp  +  d      c      c-p  +  cd      cq-\-d 
and     *.-  when  relation  (a ) holds  f or p,-"Sithold 8  forc^j9,-"C^s,[ 2 
i.e.,  for         (^p-\-cd,        •••         (?s-\-cd,  [1 

-  be  — ad  be  —  ad  r^ 

(rp-\-  cdi!  &s-\-cd 

^     a  ,   be  — ad  a  ,   be  —  ad  ri 

I.e.,  for -4--^ — ■ — -,        ..._+__-—;  [1 

c      erp-\-cd  c     c-s-\-cd 

.-.  it  holds  for  ^^^±^,    ...^^i±^,    if  fori),  ...  s.  Q.E.D. 
cp-^d         cs-\-d 


6,  7,  §  4.]  CONVERGENCE  AND  DIVERGENCE. 


§4.     CONVERGENCE  AND  DIVERGENCE. 

^^  In  this  section  all  series  are  understood  to  be  infinite, 
and  to  be  made  up  of  real,  positive  terms  only. 

The  sum  of  a  series  is  the  limit  of  the  sum  of  its  first  n  terms 
when  n  becomes  indefinitely  great. 

The  excess  of  the  sum  of  a  series  over  the  sum  of  its  first 
n  terms  is  its  remainder  after  n  terms. 

The  abbreviations  are  :  s  for  the  sum  of  the  series,  t„  for  the 

nth.  term,  s„,  for  T^  +  TaH hTft,  the  sum  of  the  first  n  terms, 

and  R„  for  T„.f i -f- T„-f.2  +  •••,  the  remainder  after  n  terms. 

An  infinite  series  is  -J   ,. .  ,  ^  ,    if  s«-J  ~  a  finite  limit; 

i.e.,  if  K^-{  ~   0,  when  n  =  oo.  ■  » 

The  terms  of  an  infinite  convergent  series  grow  smaller  and 

smaller,  since  r„,  =  Tft+i+  t„+2H j  =  0  ;  but  that  this  condition 

is  not  sufllcient  appears  from  an  example : 

In  the  harmonic  series  1  H 1 1 the  terms  grow  smaller ; 

^2      3 

but  the  series  is  not  convergent;  for  if  it  be  grouped  thus  ; 

then   •••  the  sum  of  no  group  is  less  than  ^, 
and     •.  •  the  series  consists  of  an  infinite  number  of  such  groups, 
.*.  s,j  ^  a  finite  limit  when   n  r=  oo.  q.e.d. 

Theor.  7.  Tlie  sum  of  a  convergent  series  of  positive  terms  is 
the  same  in  ivliatever  way  the  terms  are  arranged  or  grouped. 

Let  s  =  Ti4-T2+T3H ,  any  convergent  series  ;  let  the  same 

series  be  arranged  or  grouped  in  any  other  way,  say 
(t2  +  Ti)  +  (t4  +  T3)  +  ...; 
and  let  s„'  be  the  sum  of  the  first  n  groups  ; 

then  will  s,/=  s  when  71  =  oo . 
For     *.*  in  s  are  found  all  the  terms  of  s^'  and  more ; 

.-.  s„'  <  s,  and  lim  s„'  >  s.  [df.  limit 

So  s„  <  lim  s„',  and  s,  =  lim  s„,  >  lim  s„' ; 

.-.    limS  '  =  S.  Q.E.D. 


370  SERIES.  [XII.  ths. 

Theor.  8.  If  the  terms  of  a  convergent  series  he  multiplied  by 
any  same  Jinite  number,  the  new  series  thus  formed  is  convergent. 

Let  Ti  +  T2  -f-  Tg  H f-T,»  H be  any  convergent  series,  and 

k  a  constant ; 

then  is  the  series  JcTi  -\-lcT2  +  kTs-\ h  An^»  H convergent. 

For     •.•  R„,    =T„+i  4-T„+2    -f--,  =0   when  n  =  CO,  [hyp. 

.' '    JCR^',  =  ^n+l -\- ^n+2 -\ J  =  0     Whcn  W  =  00.     Q.E.D. 

Cor.  If  tJie  terms  of  a  convergent  series  be  multiplied  by  any 
finite  numbers  not  larger  than  a  given  finite  number,  the  new 
series  thus  formed  is  convergent. 

Theor.  9.  If,  after  a  given  term,  the  terms  of  a  series  form 
a  decreasing  geometric  progression,  the  series  is  convergent. 

Let  Ti  +  To  -f  T3  H 1-  Tjfc  +  r  •  Ti  4-  7^  •  T;fe  H be  a  series  such 

that  the  terms  after  a  given  term  t^  form  a  geometric 
progression  with  r  smaller  than  1  ; 
then  is  this  series  convergent. 

For     *.*  TiH \-Tj,  is  a  finite  constant  number,  s^^, 

and    '.•  T»+i+...=Tt+i.(l+rH-r2+...) 

=  Ti+i :  1  —  r,  when  w  =  00 ,  [th.  4  cr.  2 

.*.  s  =  StH — ^±^,   a  finite  number.  q.e.d. 

1  — r 

Theor.  10.  If  one  series  be  convergent,  and  if  the  terms  of 
another  series  be  not  larger  than  the  corresponding  terms  of  the 
first  series,  the  second  series  is  convergent. 

Let     ■^Ti-h+Ta+'^Tg-j }-"^T„H be  a  convergent  series, 

and  let    Ti'+  Tg'  +  T3'  +  •  •  •  +  t„'  H be  another  series  such  that 

Ti'S^Ti,      Ta'^Ts,      T3'^T3,      •.•,      T„' ^  T„,    .••; 

then  is  the  second  series  convergent. 

For       •.•    T„'+i5^T„+i,       T„V2^T„+2,     — , 
•*•    Rn   ^  K„? 

and     •.•  R„  =0,  ,  [hyp. 

.*.    R„'  =  0.  Q.E.D. 

CoR.  1.  If  one  series  be  divergent,  and  if  the  terms  of  another 
series  be  not  smaller  than  the  corresponding  terms  of  the  first 
series,  the  second  series  is  divergent. 


8-11,  §4.]  CONVERGENCE  AND  DIVERGENCE.  371 

Cor.  2.  If  one  series  he  convergent^  and  if  in  a  second  series  the 
ratio  of  each  term  to  the  term  before  it  he  not  larger  than  the  cor- 
responding ratio  in  the  first  series^  the  second  series  is  convergent. 

Theor.  11.  If  after  a  given  term,  the  ratio  of  each  term  of 
a  series  to  the  term  hefore  it  he  smaller  than  some  fixed  number 
that  is  itself  smaller  than  unity,  the  series  is  convergent. 

Let  Ti  +  Ts-l-TgH |-Ti+"«   be  a  series  such  that  after  a 

given  term  t^^  the  ratios   t^^i  :  t^,  •••    each  <  ^  <  1 ; 
then  is  the  series  convergent. 
For  form  a  new  series 

Ti  +  T2  +  T3  +  ...-|-T;,  +  t{+iH hT„'... 

identical  with  first  series  for  the  first  k  terms,  and 
thereafter  a  geometric  progression  whose  ratio  is  h  ; 
then   •.•  this  second  series  is  convergent,  [th.  9 

and     • .  •  the  terms  of  the  first  series  are  not  larger  than  the  cor- 
responding terms  of  the  second  series,  V^yV' 
.*.  the  first  series  is  convergent.                   q.e.d.     [th.  10 
Note  1.  It  is  not  sufficient  that  the  ratios  Tj^^i  :t^,"'  be  simply 
less  than  a  unit. 

E.g. ,  the  harmonic  series  1 -\ 1 j 1 .  [above 

Z  O  4: 

Note  2.  Application  of  the  theorem:  To  apply  this 
theorem,  find  the  law  of  the  ratio  t„+i  :  t„,  which  in  general  is 
some  function  of  n ;  then  determine  whether  this  ratio  r,  as  7i 
increases,  finally  becomes  and  remains  smaller  than  some  fixed 
number  7i,  that  is  itself  smaller  than  a  unit. 

If  smaller  than  h,  the  series  is  convergent. 

If  smaller  than  a  unit  simply,  there  is  doubt. 

If  a  unit  or  larger  than  a  unit,  the  series  is  divergent. 

E.g.,  ffiven  i  +  —  +  —  + h--H : 

if  '  s  1^2!      3!  n\ 

then   •.•  the  ratio  t„:  T„_i  =  -    =0   whenw  =  oo, 

n 


.'.  the  series  is  convergent. 


So,  given   l+±+L  +  ...+±  + 


372  SERIES.  [XII.  th. 

then    •.•  the  ratio  T,:T,_i=i^iqJ^'  =  (l-l)2=lwhen?i  =  x, 

w  n 

.'.  there  is  doubt. 

But,  if  the  series  be  grouped  thus  : 

i.e.,  in  groups  wherein  the  denominator  of  the  first  term  of 

each  group  is  an  integral  power  of  2  ; 

then   •.•  the  several  groups  are  less  than  -,   -,   -,  ..., 

and     *.•  the  series  l-f--  +  -  +  --j is  convergent,         [th.  9 

2      4      8 

.*.  the  first  series  is  convergent.  q.e.d.     [th.  10 

So,  given  l  +  i  +  i  +  ...: 

then    :■  s^l+(l  +  r\  +  a  +  l  +  L  +  l.) 


\8P 


<14.2    .  1      8 

a  geometric  series  whose  ratio  is  2^"*. 
.*.  8  is  convergent  when  p>l.  [th.  10 

So,  the  series  1  H 


1  + 


2  (log  2)^  ^3  (log  3)^^      ' 

1  .  1 


2  log  2  (log  log  2y  3  log  3  (log  log  3)^ 
+  •••,  and  so  on, 
I  convergent      i           >  -,  r  ^ 

^^^  ^  divergent      "^^^^"^  ^  >^'  ^^^"^"^  ^^  ^^^^® 

Note  3.     General  test  of  convergence.     The  series 

are  each  of  them  ■{  /i^^J,^^^^     when  i?  ^  1.  [ex.  above 

These  series,  when  compared  with  most  other  series,  furnish  a 
test  of  their  convergence.  [th.lO,  th.  8 


11,  §4.]  CONVERGENCE   AND   DIVERGENCE.  873 

It  is  to  be  noted  that  in  the  divergent  series    1  -\ 1 !-•••, 

I  °  2      3  ' 

the  nth  term,   -,    is   an   infinitesimal  when   71  =  00.      Let  this 

infinitesimal  be  counted  the  base  [VII.  §  4,  df .  order  of  infls.]  ; 

then   — ,    the  nth   term  of  the  convergent   series    1-1 1 — 

H )   [P>1]  is  an  infinitesimal  of  an  order  higher  than  the 

first  order  by  a  finite  number  p  —  1.  And,  conversely,  a  series 
whose  terms  are  infinitesimals  of  an  order,  p^  finitely  higher 
than  the  first  order,  is  convergent.  But  if  p>l,  and  p  =  l 
when  n  =  00,  there  is  doubt,  and  the  series  may  then  be  tested 

by  the  series  1  -\ -I 1 ,    and  so  on, 

■^  ^  2  (log  2)''  '^  3  (log  8)"^ 

i.e.,  from  the  series  -,    ,    ,  •••,  a  base 

71     ?i  •  looj  71     n  •  los:  n  •  I02:  I02:  n 

may  generally  be  chosen  for  which,  when  n  =  00,  the  order  of 
the  term  t„  of  the  series  to  be  tested  is  <j       !     ^  higher  than  the 

first  order,  and  fhe  series  is  ^  convergent^ 

'  divergent 

Note  4.  Bounds  of  error  :  In  the  summation  of  most 
series,  only  a  finite  number  of  terms  is  used,  and  only  ap- 
proximations to  the  true  value  are  found ;  and  it  is  then 
important  to  know  between  what  bounds  the  error  lies.  That 
approximation  is  s„  and  the  error  is  — R„.       [V.  § 5 df.,n finite. 

E.g.,  in  the  first  example  of  Note  2, 

and     •••  the  terms  of  this  series  are  not  greater  than  those  of 
1        ^,^^.^+       l^.^^L^,+.., 


(n+1)!       (n  +  1)!   n  +  l       (w+1)!    (^  +  1)^ 

(n  +  1)  !    \        n  +  ly         71 -nl 
i.e.,  the  error  lies  between  0  and 


7i'n ! 


In  particular,    Sq<^s< = ;  Sio  ~  s  < 


6 '6  I      4320  10.10! 


374  SERIES.  [XII.  ths. 

Note  5.     Series  arranged  to  powers  of  a  variable  :    If 
a  series  be  arranged  to  the  powers  of  some  variable  a;,  thus 

Ao  +  Aiaj  +  Agif^H hA„cc"  +  "-, 

then         the  ratio  t„+i  :  t„  =a;(A„  :  a„_i)  =  x :  (a„_i  :  a„)  , 

I  convergent  I  < 

and  the  series  is  \  in  doubt      \t  x<  ^  a„_i  :  a„,  when  w  =  oo. 
I  divergent  |  > 

E.g, ,  the  series  1-\-x-\-2\q? -\-^\3^ -\ is  divergent, 

however  small  x  may  be  ; 

for  the  ratio  a^_i  :  a„  =  1  :  n  =  0,  when  n  =  qo. 

So,  the  series  1+-+  —  +  — H is  convergent, 

however  large  x  ma}'  be. 

So   thP  aeriea  ? -i-^'j-  ^J-  ...  is  ^  convergent  :f  ^  i  <1 ; 
bo,  the  series  -  +  ^  +  ^  +  ...  is  ^  divergent     '^  ^^  <1 ; 

for  the  ratio  a^_i  :  a^  =  1  when  n  =  oo  ; 

and  this  series  is  divergent  if  a;  =  1 . 

So,  the  series  x+ 2.^  +  3^'+...  is  {  ZeTgInT* '^  =^  "!  f^. 

That  value  of  x  which  leaves  the  series  in  doubt, 
viz.,  '*'lim(A„_i :  a„)    when  n  =  oo, 

is  the  radius  of  convergence  of  the  series. 

E.g.,  if  r  =  radius  of  convergence,  then  in  the  first  of  the  ex- 
amples above  r  =  0;    in  the  second,  r=cc;   in  the 
third  and  fourth,  r=  1. 
If  some  of  the  powers  of  x  be  wanting,  the  general  method  of 
Note  2  must  be  applied. 

E.g.,the  series   1  +  ^  +  ^+ ... +i!^!^  is^  convergent 
^  '  3       7  2*"— 1         divergent 

for  the  ratio  t„+i  :  t„  =  a2(2"-l) : (2*h-1-  l) 

=  a:^ :  2    when  n  =  oo  ; 
and  the  radius  of  convergence  is  ■>/2. 


11,  12,  §  5.]  INDETEE^HNATE   SERIES.  875 

§5.     INDETEEMIiiATE    SEEIES. 

An  infinite  series  that  has  different  sums  when  its  terms  are 
arranged  or  grouped  in  different  ways  is  indeterminate. 

E.g.^  the  sum  of  the  series  +1,  ~1,  +1,  ~1,  +1,  ...  ma}-  be 
either       (1-1) +  (1_1)  +  (1_1)  + ...,  =0, 
or  14.(-1  +  1)H_(-1  +  1)  +  ...,         =1. 

Indeterminate  series,  although  not  always  divergent,  are  here 
classed  with  non-convergent  series. 

Theor.  12.    An  infinite  series  that  Jias  positive  and  negative 

terms  that  separately  form  divergent  series  is  indeterminate. 

For  take  any  positive  term  or  group  of  positive  terms  for  +Ti, 

leaving  positive  terms  whose  sum  +Ri  is  infinite, 

and  from  the  negative  terms,  whose  sum  is  infinite,  tai?:e 

enough  terms  so  that  their  sum  ~To  is  larger  than 

■*"Ti,  leaving  negative  terms  whose  sum  ^Ro  is  infinite  ; 

and  from  "^Rj  form  +T3  larger  than  "Xg,  leaving  +R3  infinite  ; 

and  from  "Kg  form  ~T4  larger  than  +T3,  leaving  ~R4  infinite  ; 

and  so  on  ; 
then    •.•  the  new  series  +Ti,  "Tg,  +T3,  ~T4,  ... 
gives      ("^Ti4-~T2)  +  (''"T3+~T4)H — ,       =  some  negative  number, 
and        +Ti + (  ^T2+"^T3) + (  T4-f +T5)  -j — ,  =  some  positive  number, 
.*.  the  series  is  indeterminate.  q.e.d. 

Note  1.  This  result  appears  also  from  this,  that  the  sum  of  the 
given  series  reduces  to  the  difference  of  the  sums  of  two  divergent 
series  and  is  of  the  form  00  —  00,  an  indeterminate  expression. 

Cor.  1 .  A  series  s  is  non-convergent  if  the  series  got  by  making 
all  the  terms  of  s  positive  be  divergent. 

For,  if  the  series  be  divergent  when  all  the  terms  are  made 
positive,  it  is  either  of  the  form  00— oo,  a— 00,  00—  a, 
when  part  of  the  terms  are  made  negative  ; 
I.e.,  it  is  either  indeterminate  or  divergent.  q.e.d. 

Note  2.  Manifestly  a  given  series  may  be  reduced  to  the  form 
+Ti,  'Tg,  +T3,  ~T4,  •••  in  an  infinite  number  of  ways,  giving  an 
infinite  number  of  such  double  values. 


376  SERIES.  [XII.  tli8. 

Note  3.  An  indeterminate  series  may  sometimes  be  arranged 
so  as  to  have  terms  alternately  positive  and  negative  and  growing 
smaller  and  smaller ;  and  if  the  terms  approach  0,  the  sum 
for  such  arrangement  has  a  single  finite  value,  but  for  different 
arranjjeiiients  different  values.  If,  for  a  particular  arrangement, 
a  series  have  a  single  finite  value,  however  grouped,  the  series 
is  convergent  for  that  arrangement. 

E.g.,  if  s  = --?  +  ---  +  ---  +  •••  towards  "^1; 
-^  1      2      3      4      5      6 

then         the  two  values  of  s  both  lie  between  0  and  2  ; 

and  s»~s„^i  =  l    when7i  =  oo. 

So,  if  s  =  1  -  i  H-i  - i  +  i  -  i  +  ...  towards  0, 
2      3      4      5      G  ' 

then         s<l,  *.•  Ri  is  negative, 

>1 ,  •.•  Rs  is  positive, 

<!—-  +  -,  •.•  Eg  is  negative, 

Z       o 

and  so  on ; 
and  Rn===0    when  n  =  oo; 


I.e. 


Y'      *^"6' 


But,  if  this  series  be  arranged  thus : 

then         s<l  +  i,    >l  +  --i,    <l  +  i_i  +  l  +  l,  ... ; 

3  3      2  3      2      5      7 

I.e.,         s<li,        >|,  <.... 

The  reader  may  group  the  positive  terms  by  threes,  or  by 
fours,  or  ...,  and  the  negative  terms  singl}',  by  twos,  or  by 
threes,  or  ...,  at  his  pleasure,  taking  care  that  the  terms  of  his 
new  series  be  always  in  descending  order  of  magnitude. 


12,13,  §5.]  INDETERMINATE   SERIES.  S77 

Theor.  13.  A  series  s  is  convergent  if  the  series  got  by  making 
all  the  terms  of  s  positive  he  convergent. 

For  let  "''s'  =  the  series  of  positive  terms  in  s, 
and  "8"=  the  series  of  negative  terms  in  s  ; 

then   •.•  +s'-h+s"  is  finite,  [^JP* 

.-.  +8',  *^s"  are  botli  finite, 

.'.  +s',  "^s"  is  the  same  however  its  terms  are  arranged 
or  grouped.  [th.7 

Let  the  terras  of  8  be  arranged  and  grouped  in  any  way, 
and  let     s„  =  sum  of  the  first  n  groups  of  that  arrangement, 
and         "^8^/  =  sum  of  the  n'  positive  terms  of  s  contained  in  s„, 
and        ~8l«=  sum  of  the  n"  negative  terms  of  8  contained  in  s„  ; 
then    •••  s„  =  8^— +8^'«, 

.-.  lim  8„  =  lim 8^,  —  lim +8^',  =  s'— +8". 

But     •••  +8',  "^s"  are  finite  constant  numbers,  [above 

.-.  8,   =+8'— ■'"s",  is  a  finite  constant  number,  q.e.d.     . 

Cor.  1.    If  a  series  b  he  {      ^      9    i  ^j^q  series  qot  hv 

^  '  non-convergent^  ^        ^ 

making  all  the  terms  of  s  positive  is-{   ■,.  ^         t  ' 

Cor.  2.  If  an  indeterminate  series  he  convergent  for  a  particu- 
lar arrangement  "•"Tj,  ~T2,  +T3,  ...  ^t^^  '''1^+1,  ...,  the  ratio  Tk+i :  t^ 
hecomes  and  remains  smaller  than  unity ^  hut  approaches  unity  as 
its  limit.  [th .  1 1  n  t.  2 

For  if  the  ratio  t^^j  :  T;^  approach  a  limit  h  smaller  than  unity, 
the  series  is  convergent  and  not  indeterminate.   [th.l3,  th.ll  nt.2 

Note.  If  indeterminate  series  be  classed  with  divergent 
series  as  above,  then,  in  the  light  of  theors.  12,  13,  it  appears 
that  theors.  7-11,  with  their  notes  and  corollaries,  apply  to  series 
with  negative  terms,  and  that  those  theorems  are  general  for 
all  series  of  real  terms. 

Indeterminate  series  are  unsafe  ;  and,  by  reason  of  their  slow 
convergence,  they  are  worthless. 


378  SERIES.  [XII.  ths. 

§  6.     IMAGINARY  SERIES. 

A  SERIES  whose  terms  are  part  or  all  imaginary  is  an  imagi- 
nary series.  If  each  term  of  the  series  Ti,  Tj,  ...  be  resolved 
into  its  two  components  Pi,  Qii;   P2,  Qa*;  ...,  the  two  series 

s',  =Pi+P2H — ,  and  s"i,  =Qii+Q2iH ,  are  the  components 

of  8,  and  s  =  s'+s"i. 

The  moduli  of  the  several  terms  taken  in  order  form  the  series 
of  moduli^  a  series  of  real  positive  numbers,    =^(Pi-  +  Qi^)"*. 

Theor.  14.    If  for  any  imaginary  series  the  series  of  moduli 
be  convergent,  the  imaginary  series  is  convergent. 
For    *.*  the  series  s',  s"  have  their  terms  when  made  positive 
not  greater  than  the  corresponding  terms  of  the  con- 
vergent series  of  moduli,  [+p  >  V(^+  Q^)  j  ••• 
.?.  s',  s"  are  convergent,  [ths.  10,13 
,♦.  s,  =s'-|-s"?',  is  convergent.                           q.e.d. 

Theor.  15.    If  for  any  imaginary  series  the  series  of  moduli 

be  divergent,  the  imaginary  series  is  non-convergent. 

For    *.*  +s'4-'*'s",  the  sum  of  the  component  series  s',  s"j,  with 

all  their  terms  made  real  and  positive,  is  not  less  than 

the  divergent  series  of  moduli,    [+p+'^Q<  VC^^+Q^) 

.*.  one  or  both  of  the  series  +s',  +s"  are  divergent, 

.*.  one  or  both  of  the  series    s',    s"  are  non-convergent, 

[th.l2cr.l 
.♦.  s,   =s'4-s"i,  is  non-convergent.  q.e.d. 

Cor.    If  she  {  ^^'^'^^^9^'^^^  so  is  its  series  of  moduli. 

-^  >  non-convergent,  -' 

Note.  Theors.  14, 15,  when  applied  to  series  of  real  numbers, 
become  theor.  13  and  its  converse,  since  the  modulus  of  a  real 
number  is  that  number  taken  positive. 

In  the  light  of  theors.  14,  15,  it  appears  that  theors.  7-11, 
with  their  notes  and  corollaries,  apply  to  series  with  imaginary 
terms,  and  that  those  theorems  are  general  for  all  series.  Theor. 
16  shows  that  every  series  to  rising  powers  of  a  variable  has  a 
radius  of  converoence. 


14-lG,  §6.]  IMAGINARY   SERIES.  379 

Theor.  16  (Abel's  theorem).  If  a  series,  Aq+AiZ  +  AsZ^-I — , 
arranged  to  rising  powers  of  a  variable  z,  be  ^  -  -      .when 

modz  =  a  constant  r,  it  is-l    ^        ^        ,.  ivhenever  mod  z  ^  ^  * 
'  '  non-convergt  '  <r. 

■c^  1  I  2  1        •    I  convergent  , 

For    •.*  Ao+Ai-2;4-A9-2;^H is<  ^  .when 

o-r  1      -T-    .       T-         -)  non-convergent 

mod  2  =  r,  [hyp 

.*.  mod  Ao  4- mod  Ai  •  mod  z  +  mod  A2 •  mod  z"  -\ is 


mod  Ao  +  mod  Ai  •  mod  z  +  mod  Ag  •  mod  z^  -\ is 

z^  >*'• 
converscent 


,  convergent     ,  !)>»'• 

<  T         °  .     whenever  mod  z<  Z^ 
'  divergent  '  <  r. 


.-.  s,  -A0  +  A12  i-Aoz'  +  "-,  is^  non-convergent 

"^  r 
whenever  mod  0  ^  ^   *  q.e.d.     [ths.  14,15 

Cor.    If  in  a  series  arranged  to  rising  powers  of  z,  mod  z 

I  convergent  \  < 

increase  from  0  to  co,  the  series  is  <  in  doubt    ivhen  modz-{  =r. 

I  divergent  \  > 

[r  a  constant,  called  the  radius  of  convergence  of  the  series. 
In  most  series  r  is  lim  ratio  mod  a„  :  mod  a„+i.  [th.  11  nt.  0 

Note.  Graphic  representation  :  Denote  by  z  the  represen- 
tative point  of  any  number  z ;  i.e.,  the  extremity  of  that  vector 
from  the  origin  whose  ratio  to  the  unit-line  is  z  ;  and  so  for  other 
numbers.  Let  Aq,  AiZ,  A2Z-, ...  be  any  series  arranged  to  rising 
powers  of  z;  and  from  o  as  centre,  with  radius  equal  to  the 
radius  of  convergence  of  the  series,  draw  a  circle  ;  this  circle, 
called  the  ciixle  of  convergence,  embraces  the  region 
I  within  I  convergent. 

<  upon      which  z  lies  when  the  series  is  \  in  doubt. 
I  without  I  divergent. 

If  a  series  be  arranged  to  rising  powers  of  (2;  —  a) ,  then  the 
circle  of  convergence  has  a  for  centre  and  r  for  radius,  and  the 

I  convergent  I  within 

series  is  -{  in  doubt     when  z  lies  <  upon       this  circle  ; 
I  divergent  |  without 

<r 
for  mod  {z  —  a)  ^  =  r. 

>r. 


380  SERIES.  [XTI.  th. 

Theok.  17.  In  a  series  arranged  to  rising  powers  of  a  variable 
z,  if  modz  be  less  than  the  radius  of  convergence  of  the  series, 
an  increment  can  be  given  to  z  so  small  that  the  increment  of  the 
series  shall  be  less  than  any  assigned  number. 

For  let  s  =  Aq-\- AiZ -\- \oZ' -\ ,  take  modz  less  than  r,  the 

radius  of  convergence,  and  to  z  give  an  increment  h 
so  small  that  mod  {z  -}-  h)  <  r ; 

then    •••  sands  +  incs,   =  \q+ Xi{z '\- h) -\- ^^{z  +  hy -\ , 

are  both  convergent  series,  [hyP- 

(z  -X-  h^'  —  Z' 

,'.  incs,  =7t(Ai4-Ao^  ' 1 ),  is  convergent ; 

.*.  inc  s  :  /i  is  a  convergent  series  when  h  is  finite  ;     [th.  8 

and     *.•  h  may  approach  0  so  that  \^{z -\- hY  —  z^~\  :  h  is  larger 

than  but  approaches  wz""',  [bin.  th. 

.*.  inc  s  :  7i  =  a  finite  limit  when  /i  =  0  ;  [th.  10 

.*.  incs,  =  /i« a  finite  number,  =  0  when  ^=0.  q.e.d. 

Cor.  1.    D^s,  =Ai4-2a2Z +3A3Z-H ,  i>z^s,  d/s,  •••,  ai^e  all 

senes  whose  common  radius  of  convergence  is  r. 

CoR.  2.   For  all  values  of  mod  z\  ■.  than  v,  the  series- 

^      ^.  ,  ,         o  ,         .     ,  a  finite  continuous  one-value 

function  A<,  +  A.z  +  A,z-  +  -  ts  J,  ^^  .^^^.^^  or  indeterminate 

function  of  z. 

If  s  be  a  series  to  rising  powers  of  a  variable  z,  and  z  be  a 
finite  function  of  z  that  is  equal  to  s  for  continuous  values  of  z 
from  0  to  r,  but  unequal  for  a  value  of  z  larger  than  r,  then  s 
and  z  are  discontinuous  when  z  -^  r  [theory  of  functions],  and 
r  is  the  radius  of  convergence  of  s  and  the  smallest  value  of  z 
for  which  z  is  discontinuous. 

In  the  graphic  representation  of  imaginaries,  if  i\iQ  points  of 
discontinuity,  a,  &,  c,  •••  of  the  function  {a~zy{b—zy{c  —  zY 
'"  (P?  ^h  '"'  •••  ^"y  fractions  or  negative  integers)  be  platted, 
and  the  function  be  equal  to  a  series  to  rising  powers  of  z  —  Tc, 
then,  with  k  as  the  centre  of  convergence,  the  radius  of  con- 
vergence of  the  series  is  the  distance  from  k  to  the  nearest 
point  of  discontinuity. 


17,  §7.]  EXPANSION  OF  FUNCTIONS.  381 

§7.   EXPANSION  OF  FUNCTIONS   IN  INFINITE  SERIES. 

If  z  =  Ti  -f-TsH-  T3  H [z,  Ti,  T2,  Tg  •••  functions  of  2;]  for  all 

values  of  z  that  make  the  second  member  a  convergent  series, 
the  series  is  an  expansion  of  z  in  functions  of  z. 

An  ordinary  function  of  a  variable  cannot,  in  general,  be  equal 
to  any  one  infinite  series  for  all  values  of  that  variable. 

E.g.^  if  2;  be  a  variable  that  increases  from  0  to  00,  then  the 
-  I  finite  and  positive  |  <  1 

fraction  is  -j  infinite  when  z\  =1 

^  —  ^       I  finite  and  negative  |  >  1 

but     *.*  the  series  is  infinite  when  2;  >  1, 

.  • .  the  series  1  -\-z-\-z^-\ — ,  which  equals  the  fraction  for  all 
values  of  z  from  0  to  1,  ceases  to  equal  it  when  2;  >  1. 

So,  the  series  —z  —  z^—'^-"^  wherein  z  =  \  :  a;,  is  an  expansion 
oi{l-x)-\=-z{l-z)-^', 
and  the  two  are  equal  when  2?  <  1 , 

i.e,  when  a;  >  1  ; 

but  the  series  is  divergent,  and  the  two  are  unequal  when  2;  >1. 

I  real  I   <  ^  ' 

So,  the  radical  V(^~  ^)  ^^  1  ^^^^  when  z  \   =  1 ; 

I  imaginary  I   >  1  I 

and  it  is  shown  later  that  an  expansion  of  y'(l  —  2;)  is 

1-i^-K-lV^--;  [bin.th. 

but  this  equality  is  impossible  when  2;  >  1  ; 

for  the  series-function  remains  real  for  all  real  values  of  2!, 

and  the  radical  becomes  imaginary  when  2;  >  1. 

o      •*       u  I  negative  integer,  .,       ,  fraction    ,  .^ 

So,  If  n  be  any  {  ^J^^.^^^  'the  ^  ^^^j^^,     (a-.)* 

may  be  expanded  into  the  series 

a«_7ia«-i2;  +  '^{^^-^)  a--'-z^ ,  [bin.  th. 

2! 
whose  radius  of  convergence  is  a ;  [th.  11,  nt.  5 

then         the  ^       ,.     ,    is  not  equal  to  the  series  when  z>a. 

So,  if  p,  g,  r  be  any  fractions  or  negative  integers,  and  if 
z  =  (a  —  zy{b  —  z)^{c  —  zy-",  then  z  cannot  equal  a 
series  to  rising  powers  of  z  when  z  is  larger  than  the 
smallest  of  the  numbers  a,  b,  c,  •••. 


882  '  SERIES.  [XII.  ths.  18,  19 

Theor.  18.  The  sum  of  an  infinite  series  Ao+AiX+AgX-^--", 
icJiose  radius  of  convergence  is  greater  than  0,  approaches  the 
limit  Ao  iL'hen  x  =  0. 

For     '.•  Ai+Ao.r-+-A3.^H —  has  the  same  radius  of  convergence 
as  the  given  series, 
.*.  it  is  convergent  for  small  values  of  x, 

.*.  the  product  x{xi-{-  a^x  -\-  a^oi?  -\ )  =  0,  when  ic=  0, 

.•.  Ao  +  Ajic -f  AgOz-^H =Aowhena;=0.  q.e.d. 

Cor.    In  the  infinite  series  a©  +  Ai  x  +  As x^  H f-  a^ x**  H , 

X  may  be  made  so  -{  .         that  a^x*"  shall  he  any  number  of  times 
larger  than  the  sum  of  all  the  terms  o/-{  j  ^ ''  !  degree. 

Theor.  19.  If  two  series,  arranged  to  Hsing  powers  of  any 
same  variable,  be  equal  for  all  values  of  the  variable  that  make 
them  both  convergent,  the  coefficients  of  like  powers  of  the  vari- 
able are  equal. 

Let  Ao  + Ai^cH-Aga^H =  Aq'+a/o;  +A2' ar^H ,  when  a;<?', 

wherein   if  the  series  have  different  radii  of  convergence,  r  is 

the  least  of  the  two  ; 
then  will  Aq  =  Ao',    Aj  =  Aj',    Aj  =  Ag',  •••. 
For     •.•  the  two  series  are  equal  when  a;  <  r,  [^JP* 

.*.  they  approach  equal  limits  when  x  =  0  ; 
i.e.,          Aq  =  Aq'.  [th.l8 

.-.  Aiif  +  Agic^H —  =  Ai'a;+A2VH —  when  a;<r. 
.*.  Ai    4-AoX  H —  =  Ai'    -^Az'x-i — whena;<?'.  [div. by  ic 
.*.  Ai  =  Aj'.  [as  above 

So        Ao  =  A2',  and  so  on.  q.e.d. 

Cor.  No  function  x  has  more  than  one  expansion  to  ascend- 
ing powers  of  a  given  variable  x. 

For  if  possible  let  there  be  two  separate  expansions  ; 
then    •. •  each  expansion  is  equal  to  x  when  it  is  convergent,  [df . 
.*.  the  two  expansions  are  equal  to  each  other  when  both 
are  convergent, 

.*.  their  coefficients  are  equal,  and  the  two  are  identical. 

Q.E.D.    [th. 


pr.4,  §8.]  UNKNOWN    COEFFICIENTS.  883 

§8.     METHOD    OF    UNKNOWN    COEFFICIENTS. 

The  method  of  unknown  coefficients  is  used  for  the  purpose  of 
changing  a  function  from  one  form  to  another.  It  consists  in 
equating  the  given  function  to  a  function  of  the  required  form 
with  unknown  coefficients,  and  then  finding  such  values  of 
these  coefficients,  if  possible,  as  shall  make  the  two  members 
laenticai.  expansion  of  fractions. 

PrOB.    4.     To   EXPAND   A   FRACTION   INTO    A   SERIES. 

Put  the  fraction  equal  to  a  serie&  arranged  to  the  rising  pow- 
ers of  some  letter  in  the  denominator  of  the  fractiori,  and  ivith 
unknown  coefficients. 

Free  the  equation  from  fractions. 

Equate  the  coefficients  of  the  like  powers  of  the  letter  of 
arrangement  in  the  two  members,  each  to  each,  and  solve  the 
equations  thus  found  for  the  unknown  coefficients.  [th.  19 

E.g.,  put -"^  .  ^    ,  =  A+Ba;  +  ca;-  +  Da;^+---; 

1— 3a;  +  5a;- 

then   •••  l  +  2a;  =  A-j-    b  ic  +    c\x- -\-    d  x^ -\ , 

-3a 


x-\-    c 

X--\-      D 

-3b 

-3c 

+  5a 

+  5b 

a=1,   b  — 3a  =  2,   c  — 3b  +  5a  =  0,  .-.; 
.-.  A=l,    b  =  2  +  3a  =  5,    c  =  3b  — 5a=  10,  ••• ; 
and  the  series  is    1 +  oa;+ 10a;-4-5a;^  — 35a;^  •••, 

wherein  every  coefficient  after  the  second  equals  three  times  the 

coefficient  next  before  less  five  times  the  one  before  that. 

RECURRING    SERIES. 

A  series  like  thart  in  the  example  above  is  a  recurring  series; 
it  is  a  compound  geometric  progression,  each  of  whose  terms  is 
the  sum  of  the  products  of  the  two  or  more  next  preceding  terms 
by  constant  multipliers.     The  group  of  multipliers  is  the  scale. 

E.g.,  c=3b  — 5a,  D  =  3c  — 5b,  e=3d  — 5c,  ••♦,  [ex.  pr.4 
and  (3,  —  5)  is  the  scale  for  the  series  of  coefficients  ; 

and  of  the  series  1,  bx,  10 x^,  bx^,  ~35a;*,  •••,  the  scale  is 

\  -\-2x 
3  a;,    5ar,  and  the  sum  is  the  fraction 


l-Zx  +  bx^ 


384  SERIES.  [XII.  prs. 

PrOB.  5.    To  FIND  THE  SCALE  AND  SUM  OF  A  RECURRING  SERIES. 

(a)  Scale  of  two  terms,  m,  u. 

Write   Tg  =  mTg  -f  ^iTj,    T4  =  niTg  -f  nxg,    T5  =  mT^  +  nxg. 
Solve  the  first  two  equations  for  m,  n,  and  test  the  values  thus 
found  by  the  third  equation. 

Write   s  =  T,(l-in)  +  T,, 
1  — m  — n 

For       •••    S  =Ti4-T2+T3  +  T4-| 

=  Ti  +  To  +  (mT2  +  ?lTi)  +  (mTg  +  riTs)  H 

=  Ti -f- To  H- m(T2  +  T3  +  T4 +•••)  + ^(Ti  +  T2  +  T3+ ...) 

=  Tj  +  T2  +  m(s  —  Ti)  +  ^s, 

.^  ^^Tjl-m)±T,^  Q.E.D.    [sol.fors 

1  —  m  —  71 
jEJ-gr.,  to  find  the  scale  and  sum  of  the  recurring  series 

1  +  Dx  -\-  lOar  +  5x^  —  Sox* '". 
Write  10ar'  =  m-5a;  +  n-l  and  5 a^  =  m- lOar^  +  w -Sec;  solve 
for  771,  n  ;  and  test  by  equation  —  35a?*  =  ?«,  •  5a^  -f-  ?i  •  lOa^  ; 
then        m  =  Sx,    n  =  —  ox^; 
and  3^1-3a:  +  5a:^       l  +  2a?      , 

l_3a;  +  5ar^      l-3a;+5a^ 

(b)  Scale  of  three  terms,  m,  n,  p. 

Write     T4=  mTg  +  UTo  +  PTi,     T5  =  mT4-|-nT3+ pTo, 
Te=mT5  +  nT4  +  pT3,    t^  =  niTg  +  nTs -FpT^. 
Solve  the  first  three  equations  for  m,  n,  p,  and  test  by  the  fourth. 

Write  s  =  T.(l-m-n)  +  T,(l-m)  +  T3. 
1  — m  —  n  —  p 

For    •••       S  =Ti  +  T2  +  T3  +  T4H 

=  Ti  4-  T2  +  Tg  +  (mTg  +  nT2  +i)Ti) 

+  (mT4  4-?^Tg+|)T2)H 

=  Ti  +  T2  +  Tg  +  m(T3  +  T4  +  T5H ) 

+  ?i(T2  +  Tg  +  T4  +  •••)  +P(Ti  +  To  +  Tg  +  ...) 

=  Ti  +  T2  +  Tg  +  m  (s  —  Ti  —  T2)  +  n  (s  —  Ti)  +i)S, 
.    g  ^ Ti(l— m  — ?i)  +  T2(l— m)  +Tg 

\—m  —  n—x)  Q.E.D. 

(c)  So,  for  scale  of  four  or  more  terms. 


5,6,  §8.] 


UNKNOWxV  COEFFICIENTS. 


!85 


EXPANSION   OF   SURDS. 
PrOB.  6.      To    EXPAND    A   SURD    INTO    A    SERIES  : 

Put  the  surd  equal  to  a  series  arranged  to  the  rising  powers 
of  some  letter  in  the  surd,  and  with  unknown  coefficients. 

Free  the  equation  from  radicals. 

Equate  the  coefficients  of  the  like  powers  of  the  letter  of 
arrangement,  each  to  each,  and  solve  the  equations  thus  found 
for  the  unknown  coefficients.  [th.l9 

E.g.,  to  expand  -^{a--\-hx)  : 

Pat  ■^{a^-\-bx)  =  A-{-BX-[-cx'^-\-T>x^-{-BX*-{-¥X^-\ ; 


then 


a2  _j_  5a:  =  A^  +  2  ABa;  4-  2  AC 


a^+2AD 
2bc 


0^3+ 2  AE 

2bd 

r.2 


«;*+. 


l2_ 


A  =  a. 


2ab  =  6,    2ac  +  b2  =  0,    2ad  +  2bc=0,  ... 


b 

IT""' 
2  a 


and 


^  a 


-Ir 
8a3' 

8a«' 


d  = 


b^ 


b^a? 


Ua' 

j5&^  , 

16  a^      128  a'' 
5 


So,V5=V(4  +  l)=2  +  l-l^  +  ^-^^^^^ 


+ 


So,  put  -^{a^-\-bx)=^A-\-BX-\-CX^-\-'D3l?-\-'E.X^-\ 


then  •.•  a''4-^a;=A^+3A^B 


iC  +  SAB^ 

+3a2c 


a,  B 


3  a' 


c= 


,-.  -^{a^-\-bx)  =  a  + 


bx 
3a2 


9a^' 
6V 


H-3a2d 

+  6  ABC 


.i^  +  3A2E 

+3ac=^ 
+3b2c 
+  6abd 


a;*+.. 


81  a^^ 
56V      106%* 


106^ 
243  a^ 


So,^/9  =  -^(84-l)-2  +  -^- 


9a^ 
1 


81  a^ 
5 


243  a^i 


12      288      20736 


Note.     This  method  of   expanding   {l-{-x)^  shows  that  a 

series  A  +  Ba;  +  ca^H exists  whose  qi\\  power  is  identically 

(1  -\-xy  ;  and  so  this  series,  when  convergent,  is  a  ^th  root  of 
(1  -\-xy.  There  are  q  such  series  corresponding  to  the  q,  qih. 
roots  of  unity.  ,  [X.  th.  ?? 


386  SERIES.  [XII.  pr. 

BE80LUTI0N   OF   FRACTIONS. 

PrOB.  7.  To  RESOLVE  INTO  A  SUM  OF  PARTIAL  FRiVCTIONS  A 
FRACTION  WHOSE  TERMS  ARE  ENTIRE  FUNCTIONS  OF  ANY  ELEMENT  : 

If  the  degree  of  the  numerator  be  not  lower  than  that  of  the 
denominator^  reduce  the  fraction  to  a  mixed  number. 

Resolve  the  denominator  of  the  fraction  into  its  prime  factors. 

Equate  the  fraction  to  a  set  of  fractio7is  foi-med  as  follows: 

For  every  prime  factor  not  repeated  write  a  fraction  tchose 
denovmiator  is  that  prime  factor;  and  for  any  pxime  factor 
repeated  k  times  write  k  fractions  whose  denominators  are  the 
firsts  second,  third,  •  •  •  kth  powers  of  the  factor. 

For  the  numerator  of  any  fraction  ivrite  an  entire  function  of 
the  given  element  with  unknown  coefficients,  and  of  degree  lower 
by  unity  than  the  prime  fictor  that  enters  into  its  denominator. 

Free  this  equation  from  fractions. 

Equate  the  coefficients  of  the  several  powers  of  the  letter  of 
aiTangement,  each  to  each,  and  solve  the  equations  thus  found  for 
tJie  unknown  coefficients  of  the  numerators. 

E.g.,  to  resolve  —^ :  [a;3-l=  (a;-l)  (ar^+aj+l) 

2/   —  1 

^^  . .         1  Aa;  +  B      ,      c 

Write  =  — ' ; 

x^-l       x'-^x  +  l       x-1 

then   *.-  l  =  (A  +  c)a:^-f-(— A-|-B  +  c)a;  —  B  +  c      [free fr. frac. 

.-.    A  +  C  =  0,      — A  +  B  +  C  =  0,      — B  +  C=l 
.-.    A  =  -^,      B  =  -|,      C  =  i, 

1  x-\-2  ,  1 


and 


x^-l          3{x^  +  x-\-l)     3{x-l) 
So,  write — ^ = \- 


(^x-l){x-2)(x-3) 
then   •••  2a^—10x  +  U 

=  A{a^-  5 X  4-  C)  +  B(a^-  4x  +  3)  -\-c(x^-Sx-{-  2), 
.'.  a+b4-c  =  2,     5a+4b4-3c=10,     6a4-3b  +  2c  =14, 

.'.    A  =  3,     B=~2,     C=  1, 

and  2a^-10a;+14        ^_3 2_+_L-. 

(x-l){x  —  2)(x-3)      x—1     a!-2     a!-3 


7,  §8.]  UNKNOWN  COEFFICIENTS.  387 

Since,  as  appears  from  Note  1  below,  the  identity 

=a(x-2){x-S)  +  b{x-1){x-S)-{'C{x-1){x-2) 
holds  true  for  every  value  of  x,  it  is  more  readily  solved  as  follows  : 
Put      x=l]  then  2 -10 -|- 14  =  a-"!  ."2,  and  a  =  3. 
Put      x=2;   then   b=-2. 
Put      x  =  S:   then   c  =  l. 


So,  write  — — ^^  = 


and 


{x-{-iy      (x+i)    {x-\-iy    {x+iy' 

then         A  =2,    b=~3,    c  =  4, 

2a^  +  a;+3^     2 3  4 

{x+iy      x-j-1    (x  +  iy    {x-{-iy 

This  fraction  may  also  be  resolved  as  follows  : 

'-'  ^i~-/^^ix2+.-  ■  ...^  [div.bya;+l 


{x  +  iy        {x+iy      (a;  +  l)^ 
2a;-l  _     2 3^ 

{x  +  iy~x  +  i    {x  +  iy 


a^d     '-'  ^~^2^Z±T-7:rtTV.^  [div.bya;+l 


2ic^-[-x-\-S         2               3,4  ,  " 

.'.  — ~ TT— = ;;H ^1    as  before. 

{x+iy     x-i-1    (x+iy    {x+iy 

So,  write  ^x'+^^-^'-^x-l  ^ _^_^bx±c_        d^  +  e 

(cc+l)(a^  +  aj  +  l)2      x+l     a^+x-hl     {x^-hx+lf 

then         4a^4-3a^-a^-4aj-l 

=  A{xF-\-x-\-iy-i- (bx+c  •  a^+x+l 4-Da;4-E)  (aJ+1) • 

Put       a;  =  —  1  ;      then   a  =  3. 

And   •.♦   {bx -{- c)  {x^  +  X  +  1) -{- Dx  + -E      [repl.  A,div.  bya;  + 1 

=  [4a;^  +  3a^-a^-4a;-l-3(a^+a;+l)T  :  (»+l) 
=  a^  — 4a^  —  6aj  — 4, 
.-.  Ba;+c=ic— 5,    Da;  +  E  =  — 2ic  +  l,       [div.bya^+a;4-l 

and       4a;^  +  3a;«-a^-4a;-l_^     3  a?-5  2a;-l 

(aj+l)(a^+a;  +  l)2         a;+l     a^+aj+l      (i«'+a;+l/ 

The  division  without  remainder  by  (x-\-l)  is  a  useful  check. 


388  SERIES.  [XII.  prs. 

Note  1.  When  unknown  coeflScients  are  got  by  giving 
special  values  to  a  variable  a;,  the  work  does  not  of  itself  show 
whether  any  development  of  the  proposed  form  be  possible, 
but  only  shows  what  the  coefficients  must  be  if  the  development 
be  possible.  That  every  fraction  is  resolvable  into  partial 
fractions  as  here  proposed  appears,  however,  as  follows. 

Let  the  given  fraction  be  — -,   wherein   u,  r,  w   are  entire 

vw 
and  prime  to  one  another ;  let  x^  be  any  value  of  a;, 
for  which  v  =  0 ;  let  Uj,  Wj  be  the  constants  that  u, 
w  become  when  the  variable  x  is  replaced  by  the  par- 
ticular value  «! ; 

then         -l,=_E!-+!lLlE:i£llZ,  =  i  +  _EL, 
vw      Wi  •  V  Wi  •  wv  V     v'w 

wherem   a  =  — ,    u'  =  -^ ;  (x—oci) ,  v'=  v  ;  (x—Xj)  ; 

Wi  Wi  V  1/5  V  i/> 

for      •.•  the  entire  expressions  WiU  — UiW,  v,  =  0  when  x  =  Xi, 

.*.  each  of  them  is  divisible  by  a;  —  Xi.  [XI.  th.  4. 

^  U'  B  U"      r  Ug'  ,,       W2-U'— Ua'-W     ,  . 

v'w     v'     v"w  |_        w  w  \        ^/ 

wherein   X2  is  an}'  value  of  x^  for  which  v' = 0,  u' = Ug',  w= Wg ;  •  •  • 


( 


Q  ,    R       ^  

=  -+—  ;     lQ  =  A-^B'X  —  Xi-{-C'X  —  Xi-X  —  X2-\ 

and  the  given  fraction  is  resolved  as  proposed. 

If  the  denominator  vw  have  three  or  more  factors,  then  one 
of  them,  say  v,  can  be  factored  again,  and  so  on. 

If  V  be  a  power  v"",  then  -  is  resolvable  by  division  into 

±  +  ^^  +  ...  +  L   ^ 

E.g.,  above,  where  v=  {x-\-  ly. 

Note  2.     One  of  the  uses  of  Prob.  7  is  in  the  integration 
of  rational  fractions : 

J  (x-l){x-2){x-3)         J  \x-l      x-2     x-Sj 
=  3  log  (a;— 1)  —2  log  (a;— 2)  -|-log(a;— 3)  +  a  constant. 


7,  8,  §  8.]  UNKNOWN  COEFFICIENTS.  889 

REVERSION    OF    SERIES. 

PrOB.  8.  If  A  VARIABLE  BE  EQUAL  TO  A  SERIES  OF  POWERS  OF 
ANOTHER  VARIABLE,  TO  FIND  THE  VALUE  OF  THE  SECOND  VARIABLE 
IN   TERMS    OF  THE   FIRST  : 

Put  the  letter  of  arrangement  of  tJie  given  series  equal  to  a 
new  series  arranged  to  powers  of  the  required  letter  of  arrange- 
ment with  unknown  coefficients^  and  in  the  neiv  series  replace 
the  new  letter  of  arrangement  by  the  given  series. 

Equate  the  coefficients  of  the  like  poiuers  of  the  old  letter  of 
arrangement^  each  to  each,  arid  solve  the  equations  thus  found 
for  the  unknown  coefficients. 

E.g.,  to  revert  the  series   y  —  ax  +  bx^  ■}- co:^ -] : 

Put      x  =  Ay-\-By^-^cy^-\ ,    and  replace  y,  y^,  y^, -"  by 

{ax-{-ba^-{-cx^-\ ),     {ax-\-boif-{-cx^-\ y,  •••  ; 


then 

• .  •  x  =  Aax  -\-  a6 
+  Ba2 

.'.  Aa=  1,    a6  4-i 

a^-\-  AC 
+  2  Bab 

+  ca^ 
ia^  =  0,    AC 

-\-2Bab  +  ca^  =  0, 

.      1                   b        ^      "^b^-ac 

and 

a         a^ 

2b'' -ac 
'        a' 

f-\-"- 

So,  to  revert  the  series  y  =  m  +  ax -\- bx' +  cx^ -] : 

then   *.•  y  —  m  =  ax-\-bx^-\-ca^+'", 

.-.  x  =  ^{y-m)-\(y-my-]-^^^^(y-my-\-.... 
a  a  a 

So,  to  revert  the  series   y  =  ax'  -\- bx'^ -\- cx^ -\ : 

then         ar  =  -2/ -f-\ — f+"-. 

a         a^  a^ 

So,  to  revert  the  series   y  —  ax  -\-  b3^  -}-  cx^  -\-  " ' : 
Put      x  =  Ay  +  Bf-{-cf -{-'•-; 
then   • .  •  x  =  Aax  +  a6  I  a^  +  ac        ic*  H — 
+  BaH     +3  Ba^b 
+  ca^ 
.-.  Aa=l,    A6  +  Ba'^  =  0,    Ac  +  3Ba2&  +  ca^  =  0, 
and  a;  =  l2/^Ay3  +  36^-ac^      ^,,^ 


390 


SERIES. 


[XII.  ths. 


§9.     BINOMIAL    THEOREM. 
The  OR.  20.   Ifa.-\-hbe  any  binomial,  and  n  any  real  number, 
then  (a -fb)"=a°  +  na"-^b  +  5i5_Zll)a»-2b2  _!_... 

r ! 
For,  put  x  =  b'.a', 

then         (a4-&)"  =  «"(!  +  bTaY=  a"(l  +  xy, 
(a)  n  commensurable. 

(l4-a;)''=l+Ba;H-ca;^+Da^H — ;  [B,c,D,»'»unkn.,pr.6nt. 
n(l+a;)— ^=B-f2ca;+3Dar^+...,  [Vll.th.  17,cr.l; 


Put 
then 
and 


ri(l  +  aj)"  =B  +  2c 
4-    B 


+  2c 


a^  + 


[mult,  b}^  1  +  a; 


But 


n(l  +3^)"  =  n  +  riBo;  +  wca^  4- woaJ*  + 


[above 


B-f-2c 
+    B 


a;-f-3D 
4-2c 


a^_j =  7^^-?^Ba;^-nca:^^-7^Da^+ 


and 


and 


B  =  7l,  2C  +  B  =  71B,  3 D  H- 2 C  =  ?IC, 
B 


w(n-l)  n(?i-l)(n-2) 

-    ,         ,  ri(?i  — 1)    9  ,   n(n  — 1)  (n  — 2)    o  , 
=  I  +  y^a;  +    ^  ^  ar  +  -^^ -^ ^af+..., 

(a+  6)«=  a"  +  Tia^-^d  +  ^^^^^^a^-^ft^  4. ....  q.e.d. 

(6)  n  incommensurable,  a  case  of  limits. 

Note.     Although  the  form  of  the  series  does  not  depend  on 
the  ratio  b :  a,  yet  the  series  is  worthless  unless  convergent. 

E.g.,    V5  =  V(4  +  l)  =  4^  +  i-4-^-4-4-^  +  iV-4-^--' 
and  the  convergents  are  2,  2\,  2|^{,  24|i,  ••• ; 

but  V'^=V(l+4)  =  l+i-4-i-4'  +  TV-^'-'  •••» 

and  the  convergents  are  1,  3,  I,  5,  •••,  which  are  useless. 

So,       V3=V(4-1)  =  2,  If,  m,  IIM,  ...; 
but  y— 3=V(1— 4)=  1, -1,  -3, -7,  •••,  which  is  absurd. 


20,21,  §9.]  BINOMIAL  THEOEEM.  391 

Theor.  2 1 .     The  series  1  +  n  +  ^  ^"  ~  "^^  H 

p(n-l).--(ii-r  +  l)    ,  n(n-l)-..(n-r+l)(n-r) 
r!  "^  (r  +  l)! 

is  convergent  if  n  he  ^oositive. 

1.  r  may  he  made  so  large  that  t^^o  '  (r  +  1)°"'"^<  t^+i  •  r"+^. 
For  [T.,, .  (r  +  1 )»«]  :  (t,^,  • ,-+' )  =  (^-±^Y '  (fTt) 

and  -  may  be  taken  so  small  that  A<1.  q.e.d.  [th.  18,  cr. 

r 

2.  TJie  series  is  convergent. 

For  •••  after  r  becomes  larger  than  some  fixed  finite  value  r\ 
each  product  t,^i  •  r**"^^  is  smaller  than  the  product 
before  it,  [1 

and     •••  Ty/+i-r''*+^  is  some  finite  number,  say  A;, 

.*.  the  series  is  convergent.  q.e.d.     [th.  11,  nt.  3 

Cor.  Tlie  expansion  of  (a  +  a)°  is{  '°Z7onv!Tgent  '^  °  *' 
<^^i;  anatnatofi.±,Yisi  ZrjZr,.nt  if^<  > 

Note.  The  expansion  of  (a-\-aY  is  indeterminate  if  n  He 
between  0  and~l;  for  then  the  successive  terms  of  the  series  do 
not  grow  larger,  and  are  alternately  positive  and  negative. 

But  if  the  negative  terms  be  made  positive,  the  series  is  the 
expansion  of  (a  — ay,  whose  value,  a  negative  power  of  0,  is 
infinite. 

The  expansion  of  (a-^-d)"*  is  divergent  if  w  <  —  1  ;  for  then 
the  successive  terms  of  the  series  grow  larger  and  larger. 


392  SERIES.  [XII.  pr.  9, 

PrOB.  9.       To    EXPAND    A   POWER   OF    A    BINOMIAL  : 

Reduce  the  given  expression  to  the  type-form   (a  +  b)°  and 
apply  the  binomial  formula.  [tli.  20 

E.g.,    {x-yy=^+i^{-y)  +  ^o?{-yy 

=  rc*  —  4  a^y  +  6  a^2/^— 4  ic^  +  2/^. 

So,        (2a-36)-*=  (2a)-''+-4.(2a)-'^.(-36) 

.  -4.-5 


2! 
-4.-5.-6...-(r+3) 


(2a)-«.(-36)2 


+ ...  +  ^    ^    ^      y-^^f  .2a-'<'^'^ .  (_ 36)'- -f  .. 
(2a)*       {2ay        2!.(2a)«         3!.(2a)^ 
4.5.6...-(r+3)(36y 

So,        (0^  +  2^)^  =  05^ +i.a;-^2/  +  |-~|~a.-V 

,1         -1         -3       1        -5     o     , 

2        2        2  3! 

1-1     -3     -5      3-2r    1    -?r_Li  ^  , 


=  a;^  +  ia;"*2/ - -«~V  +  — ic"  V- -^a;"V 
2       ^8       ^16  128      ^ 

,  1.3.5.7...(2r-3)    -H^izI  ^ 
+  ...  ± ^ Lx     2  yr^:..,. 

Note.  If  n  be  a  positive  integer,  the  series  ends  with  the 
(n+l)th  term,  since  the  coeflScients  of  the  following  terms 
become  0  ;  but  if  n  be  a  negative  integer,  or  a  fraction,  positive 
or  negative,  the  series  does  not  end,  and  is  infinite. 


th.  22,  §  10.] 


FINITE  DIFFEEENCES. 


393 


§10.  FINITE  DIFFEKENCES. 
If  there  be  any  series  of  numbers,  and  if  a  second  series  be 
formed  by  subtracting  each  term'  of  the  first  series  from  that 
which  follows  it,  in  order ;  a  third  series,  by  subtracting  each 
term  of  the  second  series  from  that  which  follows  it,  and  so  on ; 
then  the  terms  of  the  second  series  are  called  the  differences  of 
the  first  order,  or  fij'st  differences;  the  terms  of  the  third  series 
are  tJie  differences  of  the  second  order,  or  second  differences; 
and  so  on. 


E.g.,ifl, 

4, 

9,  16 

,  25,  36, 

•  •  be  a  series, 

then         3, 

5, 

7,    9 

11,13, 

•  •  are  the  first  differences. 

2, 

2, 

2,    2 

2,    2, 

••  are  the  second  differences, 

0, 

0, 

0,    0 

0,    0, 

••  are  the  third  differences. 

So,  if  a. 

&, 

c, 

d,      e. 

•  ••be  any  series, 

then         «!, 

&i, 

Ci, 

(Xj,       gj. 

•  •  are  the  first  differences. 

a2, 

K 

C2, 

C^2?        ^2, 

•  •  are  the  second  differences, 

as, 

h. 

Cs, 

C^3.       ^3,          . 

•  •  are  the  third  differences, 

and  so  on : 

wherein   ai= 

--h- 

-a. 

6j=c  —6, 

Ci=d—c,    •••, 

a2= 

-h- 

-«!, 

62=  Ci—  61, 

C2=di— Ci,    •-, 

and  so  on. 

The  series  a,  «!,  ag,  ag,  •••  is  the  auxiliary  series;  and  the 
object  of  the  theorems  that  follow  is  to  show  how  to  find  one 
term  and  the  sum  of  any  number  of  terms  of  the  principal  series, 
by  aid  of  the  auxiliary  series. 

Theor.  22.  If  a,  b,  c,  d,  e,  •••  he  any  series,  and  a,  ai, 
as,  83,  •••  its  auxiliary  series,  and  if  t^  he  the  {n  +  l)th  term 
of  the  principal  series;  then 

t^  =  a  +  nai  -f5-l^zJia2+  •••  +  na^.i^  a^. 

1 .    The  law  is  true  when   n  =  1 ,  2,  and  3. 
For     •••  a^—h  —a,  [df. 

.•.  6=a  +  ai.  Q.E.D.     [71  =  1 


394  SERIES.  [XII.  ths. 

So,       bi^a^-t-ao,    c  =  64-6i,   d  =  c  +  Ci, 
.-.  c,=  b-hbi, 

=  a4-2ai  +  a2.  q.e.d.     [n=:2 

So,       Ci  =  ai  + 2^2  +  03, 
.-.  d,=  c  +Ci, 

=  a -f  Sai  +  Sas  +  ag.  q.e.d.     [n  =  3 

2.  If  the  laio  he  true  when  n  =  k,   it  is  true  when  n  =  k  +  l« 
For,  let  g,  r  be  the  (^*-f  l)th  and  (A;4-2)th  terms  of   the 

principal  series, 

then   *.•  q  =  a-\-Cik'ai-\-Q2lC'a<i-\ f-c^-a^H f-a;k,  [hyp. 

and         yi=«i  +  Ci^*-aoH hc^-i^*-«rH V^i^'O^k  +  cLk+ii 

=  a  +  Ci(A:+ 1)  •  ai  +  c^ik-^-l)  •  03  H he,  (fc+l)  •  a^ 

H l-Ci(A;-f  l).a;k4-a*fi.      Q.e.d.  [IV.  th.3,cr.2 

3.  T^e  law  is  true  universally. 

For     *.*  it  is  true  when  n  =  3,  [1 

.*.  it  is  true  when  n  =  3  +  1  =  4,  [2 

.'.  it  is  trae  when  w  =  4  -f  1  =  5, 
.*.  it  is  true  when  n  =  6,  7,  8,  •••.  q.e.d. 

Note  1.  The  reader  may  compare  this  proof  with  the  third 
proof  of  the  binomial  theorem.  [V.  th.  1,  nt.  2 

Note  2.  This  theorem  is  of  special  value  when  the  auxiliary 
series  is  short,  ending  in  zeros. 

E.g.,  of  the  series  1,  8,  27,  64,  125,  •••,  the  auxiliary  series 
is  1,  7,  12,  6,  0,  0,  ..•; 

and  Tio  =  1  +  9  .  7  +  —  .  12  +  ^-^^^  -  6  =  1000. 

2  6 

So,  of  the  series  7,  16,  27,  40,  55,  •••,  the  auxiliary  series 
is  7,  9,  2,  0,  0,  ...  ; 

and  Tio=7  +  9. 9  +  ^-2  =  160, 

T„  =7 +  (71-1)9 +1(71-1)  (72 -2)2  =  n(n  + 6). 


22-24,  §10.]  FINITE  DIFFERENCES.  395 

Theor.  23.  If  the  terms  of  a  series  be  like  entire  functions 
of  their  number  in  the  series^  the  auxiliary  series  ends  with  that 
term  whose  number  is  one  greater  than  the  degree  of  the  function. 

Let  the  general  term  of  the  series  be 

T„,  =  A  +  BTi  -f  cn^  -\ 4-  Kn"* :  [ma  pos.  integer 

then   the  general  term   t^'   of   the   series   of   first  differences 
«i5  ^^  Ci,  ••.,  is 


T„+i-T„,  =  B  +  c(n-|-l  — n2)-f  ...-f  k(71  +  1   -  n"') 

=  B+c(27i-f  1)H hK(m.?i"'-i-+--")? 

which  contains  no  higher  power  of  n  than  n"*~^ 

So,  the  general  term  t^"  of  the  series  of  second  differences 
a2,  bz,  Co,  ••',  is  T„^i'  — T„',  and  contains  no  higher 
power  of  n  than  n*""^;  •••.    . 

So,  the  general  term  of  the  series  a^_i,  6„_i,  c„_i,  •••, 
contains  only  the  first  power  of  m. 

So,       the  general  term  of  the  series  a«,  6^,  c„,  •••,  is  free 
from  m,  i.e.,  is  constant, 
and  all  the  subsequent  series,  a^+i,  b^^i,  •••,  ••♦,  consist 

of  zeros.  q.e.d. 

Theor.  24.  If  the  terms  of  a  series  be  like  entire  functions  of 
their  number  in  the  series,  the  form  of  these  functions  is  identical 
with  that  found  by  aid  of  the  auxiliary  series. 

For     •.*  A  +  B7i-f  CTi^H hKn"* 

,  ,        ,,       ,   (n-l)(n-2) 
=  a  +  {n-l)ai-{-^ ^^ ^-a^-^'-- 

^  (n-.l)(n-2)...(n-m)^^ 
m ! 
for  all  integral  values  of  w,  [ths.  22,  23 

.*.  these  two  functions,  each  an  entire  function  of  n  of 
the  mth  degree,  are  equal  for  more  than  7n  values 
of  the  variable  w, 
.*.  they  are  identical.  q.e.d.     [XI.  th.  4,  cr.  3 


396  SERlEi  [XII.  th.  25. 

TiiEOR.  25.  7/"  a,  b,  c  •••  1  he  d  terms  of  any  series,  and 
a,  Ri,  Oo,  ag  •••  t^s  auxiliary  series,  then 

I  u  I       I         II  I  n(n— 1)       ,  n(n— 1)  (n--2)       , 

2  1  o  I 

For,  from  the  given  series  form  a  new  series, 

0,  a,  a  +  6,  a-h^  +  c,  •••,  a-^b-\-c-^ f-^ 

wherein  the  (n  -|- 1 )  th  term  is  the  sum  of  the  first  n  terms  of  the 

given  series ; 
then   *.•  the  nrst  differences  of  the  new  series  are  the  terms  of 
the  given  series, 
the  second  differences  of  the  new  series  are  the  first 

differences  of  the  given  series  ; 
and  so  on ; 
.*.  the  auxiliary  series  of  the  new  series  is  0,  a,  ai,  a^,  Oa,  •••, 
and  its  (w  +  l)th  term  is 

Q.E.D.      [th.22 
§11.     INTERPOLATION. 

If  for  a  series  of  values  of  a  variable  (the  arguments)  there 
be  a  corresponding  series  of  values  of  some  function  of  that 
variable,  the  insertion  of  intermediate  values  of  the  function  cor- 
responding to  intermediate  values  of  the  variable  is  interpolation. 

PrOB.  10,  To  INTERPOLATE  VALUES  BETWEEN  THE  TERMS  OP 
A   GIVEN    SERIES. 

(a)  The  form  of  the  function  known:  Apply  the  law  of  forma- 
tion, as  shown  by  the  form  of  the  function  of  n. 

E.g.,    of  the   series   1,  4,  9,   16,   ••-,    the    (2|^)th   term  is 

(2o^  =  6J. 

So,       of  the   series  1,  4,  7,  10,  13,  ••..   the   (3J)th  term  is 

7+i-3,  =  8. 
So,       of   the   series    1,  1,  16,  84,  ••.,    the    (3J)th   term   is 

1  .  4%  =  32. 


pr.  10,  §  11.]  INTERPOLATION.  397 

(6)  The  arguments  equidifferent^  and  the  form  of  the  function 
unknown : 

From  the  given  series  form  the  auxiliary  series,  and  find  the 
nth  term  of  the  given  series  by  aid  of  the  formida  of  theor.  22. 

Assume  the  law  of  formation  to  he  that  shown  in  the  form  of 
the  nth  term^  and  get  intermediate  terms  by  the  application  of 
this  law^  as  in  case  (a) . 

E.g.^    of   the   series    1,  3,  6,  10,  15,  21,  ••♦,   the  auxiliary 
series  is    1,  2,  1,  0  ; 
the  Tith  term  is 

H-2  (72- 1) +  i  (n  -  1)  (n  -  2),  =  in  (71  +  1)  ; 
and  the  (2^)th  term  is  ^^^' ^,  =  4f . 
So,       of  the  series    1,   1.414,   1.732,  2,  2.236,   2.450,    the 
values  of  the  square  roots  of  1,  2,  3,  4,  5,  6,  correct 
to  three  decimal  places,  the  auxiliary  series  is 
1,  .414,  -.096,  .046,  -.028,  .02,'...; 
and  the  approximate  value  of  V^i  ^^ 

1  -f  f  .414  -  f  .096  -  tV  .046  -  -^  .028  =  1 .581. 
Note  1.  This  rule  assumes  that  the  law  of  formation  of  the 
series  is  that  found  by  aid  of  the  auxiliary  series  and  the 
formula  of  theor.  22,  The  right  to  make  this  assumption 
appears  as  follows :  if  the  auxiliary  series  terminates,  the  for- 
mula gives  a  law  of  formation  by  which  the  integral  terms  may 
be  found,  viz.^  that  the  function  be  a  rational  integral  function 
of  the  argument;  and,  since  the  function  so  found  is  a  con- 
tinuous function,  by  its  aid  intermediate  terms  may  be  got. 

Whether  the  original  series  was  got  by  this  law  does  not 
appear;  but  as  this  is  the  simplest  law  made  known  by  the 
data,  and  as  this  law  does  give  the  integral  terms,  it  is  as- 
sumed as  the  law  of  formation  of  intermediate  terms. 

If  the  given  series  consist  of  two  terms  «,  6,  then  the 
auxiliary  series  is  a,  aj,  and  the  formula  of  interpolation  for 
Tn+i  is  a-{-7iai,  the  ordinary  formula  of  proportional  parts 
in  common  use  with  arithmetical  tables. 

E.g.,    log 500     =2.6990,  log 501  =  2.6998, 

and  log  500.6  =  2.6990  +  .6  X  .0008  =  2.6995. 


398  SERIES.  [XII.  pr. 

If  the  given  series  consist  of  three  terms,  «,  6,  c,  then  the 
auxiliary  series  is  a,  Oj,  (Tg,  and  the  formula  of   interpolation 

for  T„+i  is 

,          ,  ?i  (71—  1) 
a  +  nai-f-— ^^-^ -^«2. 

E.g.,    log  150  =  2.1 761,  log  160  =  2.2041,  log  170  =  2.2304  ; 

then  the  auxiliary  series  is  2.1761,    .0280,    —  .0017, 

and  log  163  =  2.1761  +  1.3  X  .0280  -  h^-A  x  .0017, 

=  2.2122. 

So,  if  the  given  series  have  but  four  terms,  five  terms,  and 
so  on. 

If  the  given  series  be  infinite,  the  formula  of  interpolation 
is  also  infinite,  and  it  is  available  when  convergent,  i.e.,  when 
no  term  of  the  auxiliary  series  a,  ai,  a^^  •••  exceeds  a  given  finite 

limit ;  for  since  the  series  of  coefficients  1  +  ?i  +  ^^^^  ~ — '-  -j 

is  convergent  when  n  is  positive  [th.  21],  so  is  the  formula  of 
interpolation  a  -f-  na^  +  ^n{n  —  1)02-^  '"  convergent,  [th.  8,  cr. 

When  available,  this  formula  is  better  adapted  to  computation 
than  is  the  more  general  formula  of  -case  (c) . 

(c)   The  argument's  not  equidifferent : 

Let  Xi,  Xo,  X3,  •••  x,^i  be  any  arguments  not  equidistant^  and 
yii  y2i  Jsi  ••*  Jm  t^^  corresponding  values  of  the  function,  to  in- 
terpolate a  value  of  the  function  y,  corresponding  to  a  given 
argument  x  ;  compute  y  hy  the  formula 

y  ^  (a? -  x^)  {x-x^)  —  {x- a?^^i) 

^    {x-x^){x-x^)  —  {x-x^^^)y^ 
{x^—x^)  {x^-x^)  ...  {x^-x^-^) 


{X  —  X^)  (X  —  X2)'"{X  —  X^) 

ifm+V 


{^m+i-^i)  (a^«.+i-a?2) ...  (a;„+i-a;J 


10,  §  11.]  INTERPOLATION.  899 

For,  assume  y  to  be  an  entire  function  of  x  of  the  mth  degree, 
and  write 

2/  =  A  +  bo;  -f-  ca;2  H f-  mx"*, 

then  is  this  function  identical  with  that  written  above. 

For     •••  they  are  equal  when  x=Xi^  when  x=X2^  when  x—x^j^^^ 

i.e.,  for  m  H- 1  values  of  a;, 

.-.  they  are  identical.  [XI.  th.4,  cr. 

E.g.,         if  aji,  x^,  x^=zlbO,  160,  180, 
and  2/1^2/2,2/3  =  2.1761,  2.2041,  2.2553, 

to  interpolate  a  value  of  y  corresponding  to  .7;=  163  : 

then         y  =  -^lllL  2.1761  +  15llll2.2041  +  iM_2.2553 
-10.-30  10.-20  30-20 

=  2.2122. 

Note  2.     When  iCi,  X2,  •..,  have  a  constant  difference  1,  the 
formula  of  case  (c)  is  equivalent  to  that  of  case  (6) . 
For     *.*  each  of  these  formulae  makes  y  an  entire  function  of 

X  of  the  mth  degree*, 
and     •••  both  formulae  give  the  same  value  to  y  for  rnore  than 
m  values  of  x, 
.*.  the  two  functions  are  identical.  [XI.  th.4,  cr. 

Note  3.  The  principle  of  interpolation's  illustrated  graphi- 
cally in  the  platting  of  curves  by  means  of  points.  The 
abscissas  of  the  points  represent  arguments  ;  the  corresponding 
ordinates  represent  the  known  values  of  the  function ;  and  any 
intermediate  ordinate  represents  an  intermediate  value  of  the 
function.  Graphically  the  interpolation  is  effected  hy  joining 
the  given  points  by  the  simplest  smooth  curve  that  can  be 
drawn  through  them,  and  measuring  the  ordinate  that  corre- 
sponds to  any  given  argument.  The  most  reliable  part  of  this 
curve  is  commonly  that  which  is  not  too  near  either  end. 


400  SEEIES.  [XII.  ths. 

§  12.   Taylor's  theobem. 

Lemma.  If  f{x-\-y)  be  any  finite  continuous  function  of  the 
sum  X  -h  y  for  all  values  of  that  sum  between  a  and  b,  then  for 
all  such  values    D^f  (x  -|-  y)  =  D^f  (x  +  y) . 

For  if  X  be  increased  by  h  while  y  stands  fast, 

then         Dj{x  -\-y)  =  lhn-^(^  •*" -^  "^  ^'^  ~-^^^  +  ^^ ,'     [df .  deriv. 

and  if  y  be  increased  by  h  while  x  stands  fast, 

then         V(x  +  3,)  =  lim^a£+l+lbl.^i£+ll. 

.-.  JyJ{x-\-y)=Dj{x-\-y).  [II.  ax.  1 

Theor.  26.  If  t{x-\-y)  be  a  continuous  function  of  the  sum 
(x  -f-  y)  thai  does  not  become  infinite  when  y  =  0,  its  expansion 
in  powers  ofy  can  contain  no  negative  powers  ofy. 

For  if  possible  let  the  expansion  contain  a  term  cy'*^, 
wherein   c  is  independent  of  y  ; 
then   *.•  c^~'*  =  oo    when    y=0, 

.-.  /(aJ  +  2/)  =  oo, 
which  is  contrary  to  the  hypothesis, 

.*.  this  expansion  can  contain  no  negative  powers  of  y, 

Theor.  27.  Ift(K-\-y)  and  its  successive  derivatives  be  finite 
and  continuous  functions  of  the  sum  {x-\-y),  the  expansion  of 
f  (x  +  y)  can  contain  no  fractional  power  of  y. 

For  if  possible  let  the  expansion  contain  a  term  cy'^^q, 

wherein   c  is  free  from  ?/,  w  is  a  positive  integer,  and  ^  is  a 

proper  fraction ;  ^ 

then         the  (w  +  l)th  derivative  of  this  term  as  to  2/  is  c  y?"^, 
wherein   c'  is  free  from  y,  and  - —  1  is  negative, 

and     •.*  c'yq~'^  =  'x>    when   2/==0, 

.-. /('*+i)(a;  +  y)  =  oo, 
which  is  contrary  to  the  hypothesis ; 

.-.  this  expansion  can  contain  no  fractional  powers  of  y. 


26-28,  §  12.]  Taylor's  theorem.  401 

Note.  It  is  shown  in  the  theory  of  functions  that  if  a  func- 
tion of  y  and  its  ^/-derivatives  be  finite,  continuous,  and  one- 
valued  for  all  values  of  y  smaller  than  a  constant  r,  the  function 
may  be  expanded  to  a  series  of  rising  integral  powers  of  y  that 
is  convergent  when  y  is  smaller  than  r.  This  is  equivalent  to 
sajing  that  f{x  +  y)  may  be  expanded  to  rising  integral  powers 
of  y  when  f{x-\-y)^  /'(^  +  2/)?  •••  are  finite,  continuous,  and 
one-valued  functions  oi  x-{-y  from  2/  =  0  to  2/  =  ?'• 

Theor.  28.  (Taylor's  Theorem.)  If  t{x-{-y)  be  continuous, 
and  if  it  be  possible  to  expand  this  function  in  a  series  to  positive 
integral  powers  of  y,  then 

f(x  +  y)  =  fx  +  2fx+2!f"x  +  |^f"'x+...4-^f'^)x-f..., 

wherein  fx,  f'x,  f  "x,  f '"x  ...  f^°^x  ...  are  what  f(x-f-y)  and  its 
successive  derivatives  become  ichen  y  =  0. 

For,  put/(aj  +  2/)  =  A  +  B?/  +  C2/^  +  D?/3H j-K?/°4----, 

wherein  a,  b,  c,  d,  ...  k,  ...  are  finite  and  continuous  functions 
of  ic,  but  free  from  y,  and  whose  first  derivatives  as 
to  X  are  all  finite  ;  [t^J'P* 

then         A  =fx ; 

and     •.•  D,/(a;  +  ?/)  =  D,A  +  D,B.2/  +  D,c.?/2  +  ----fi>xK2/'*+--, 

and  D,/(a;  +  2/)  =  0  +  B  +  2c2/+3D2/2  +  ...+7iK2/-i+--, 

and     •.•  T)J{x  +  y)  =  'Dyf{x-{-y),  [lem. 

.-.  d^a  +  d^b-2/  +  i>xC-2/^+---  =  b  +  2c.?/+3d2/^+--- 
for  all  finite  values  of  y,  [th.l8,  cr.l 

.-.  B  =  D,A=/'a;, 

2c  =  D,B=/"aj,         .-.  c  =  i/"a;; 

3d=d^c  =|^/"'a;,     .-.  d  =  — f'x...-,  and  so  on. 

o  ! 

.-.  f(x-{-y)=fx  +  ^f'x  +  ff"x  +  ^f"'x  +  -''.      Q.E.D. 
1  Ji  I  o  I 


402  SERIES.  [XILths. 

APPLICATIONS   OF   TAYLOR's   THEOREM.  , 

B}'  the  usre  of  Taylor's  formula  aew  methods,  often  simple,  are 
found  for  the  expansion  of  many  expressions  in  series. 

1.  Tlie  binomial  formula. 
l.Qtf{x  +  y)  =  {x  +  yY; 

then   •••  fx  =  x'\  /'x  =  na;'*~^,  /"a;  =  n(n— l)a;'*~^,  ..., 

[Vlll.ths.  16, 14 

.-.    (^x  +  yY  =  x^  +  n:^-^'y-\-''^'\~^'>^-Y^-"-, 
a  convergent  series  when  x^y.  rth.  10,  ut.  2 

2.  Tlie  logarithmic  series. 
Let  f{x  -{-y)  =  log^(a;  +  y)  ; 

then   •.•  fx  —  \ogj,x^  fx  =  M^a;~^  f'x  =  —  M^a;~^,  /'"a;  =  2 M^a;"^, 

.•.log.(x  +  ,)  =  log.x  +  M.(|-jJ  +  ^-5+...). 

CoR.    7/x=l,iAenlog^(H-y)  =  MYy-|'  +  |'-^+-^; 
a  convergent  series  when  2/  <  1 ; 

2  3  4 

So,  wheuy  <1,  log.,(l_2/)  =  M^(-?/-'^"-|--.^ ). 

2       o       4 

3.  Maclaurin' s  formula. 

Let  fy,  /O,  /'O,  /"O,  ...  be  what  f{x  +  y),  fx.fx,  f'%  -. 
become  when  a;  is  0  ; 

then        /^=/0+/'0.2/+-^-2/^+-^-2/^+-... 

4.  The  exponential  seHes.     ' 
Let  /2/  =  A"  ; 

then   .•./0  =  A«=1,    /'0  =  a«:m^=:miS  /"O  =  a«  :  m^^  ^  mjS 

...  ..=  l  +  M-.2/  +  ^  +  ^V-, 
wherein   m^S  mj^,  ...,  =:logeA,   (log.A)^,  .... 

CoR.  1.    ey=l+v  +  ^H-^+.--; 
2  !      3  ! 

a  convergent  series  for  all  finite  values  of  y^  [th.  11,  nt.  2 

and  e  =  ei=  1  +  1 +  —  +  —  +  •••  =  2.7182818-.. 

2!  3!  [IX.  th. 


28, 29,  §  13.]   COMPUTATION  OF  LOGAEITHMS.         403 

§  13.  COMPUTATION  OF  LOGARITHMS. 

Theor.  29.    Ifjube  ayiy  number  greater  than  1,  a  any  positive 
base^  and  m^  the  modulus  of  the  system;  then 

l0g^N  =  l0g^,(N-l) 


2n-1      3(2n-1)^      5(2n-1)' 

For  take   y  = ,    whence  -^  = ; 

^      2n-1  1-2/     N-1 

then    •.•  log,(l+2/)  =  M,(2/--J  +  -f-^  +  -)- 


and 


log.(l-2/)  =  M,(-^-|-^-^...).  [th.28ap.2 


...  log,i±^  =  2M.(2/+^  +^  +  1  +  ...).  [IX.th.6 

i—y  o      a       7 

and     •.•  log^N  -  log,,(N  -  1)  =  log^— ^  =  log^-^ ; 
^  N  —  1  i  — 2/ 

.         .-.    l0g^N  =  l0g,(N-l)  +  2M,(2/  +  ^  +  ^  +  ^  +  -) 

==log.(.-l)  +  2M.(^+^^^3H-..) 

This  series  is  convergent  if  n  >1.  [th.  11  nt.4 

Cor.  1.    If  ^ —1  be  any  2^osUive  fraction,  however  small, 
then  %.(N -1)  =  %.N  -  2m.(^^  +  ^^^^_^y  +  •••)• 


Cor.  2.    If  a  =  e,  the  Napierian  base,  then  m^  =  1, 

^2n-1      3(2n-1)^ 


and  Zc>(/eN  =  %e(N  — 1)  +  2(^/    .+^^,^J     ^xa  +  '")' 


404 


SERIES. 


[XII.  prs.  10, 11. 


PrOB.  11.     To   COMPUTE   A   TABLE   OF  NAPIERIAN  LOGARITHMS: 

Beginning  with  2,  compute  the  loganthm  of  every  prime  num- 
ber in  order.  [th.  29  cr.  2 

For  the  logarithms  of  composite  numbers^  add  together  the  log- 
arithms of  their  factors.  [IX.  th.  6 

^.9'olog.2=log,l  +  2Q 


3      3.33  5.3^     7-3' 

logl         =0 

.G66G6007  :  1  =  .GGGG66Q7 

7407407:3  =  24(;i)13G 

823045:5    =  104009 

91449:  7    =  13004 

10101 :9    =  1129 

1129:11=  103 

125:13=  10 

14:15=  1 


=  .693147 

So,  log,3  =  log,2  +  2  A +  -l:-3  +  J:^^+...')  =  1.098612. 
So,  log, 4  =  2. log, 2  =1.386294. 

So,  log,5  =  log.4  +  2g  +  ^  +  ^ +...)  =  1.609438. 
So,  log,  10  =  log,  2  + log,  5  =  2.302585. 

PrOB.  12.  To  COMPUTE  A  TABLE  OF  COMMON  LOGARITHMS: 

For  prime  numbers,  multiply  the  Napierian  logarithms,  found 
as  above,  by  .43429448. 

For  composite  numbers,  add  the  logarithms  of  their  factors. 
For,     logio  N  =  log, N  :  log,  10,  [IX.  th.  8 

=  log,N:  2.302585 
=  log,N  X  .43429448, 
wherein        Mjo,  =    .43429448,  is  the  reciprocal  of  2.302585. 

E.g.,    Iogio2  =    .693147  X  .43429448  =  .301030.  [pr.  9 

So,  logio3  =  1.098612  x  .43429448  =  .477121. 

Note.    The  work  is  further  shortened  bj'  interpolation. 


§  14.]  EXAMPLES.  405 

§  14.    EXAMPLES. 

§1- 
•  ••  4.    Find  the  last  term  and  the  sum  of  5  terms,  20  terms,  35 
terms,  50  terms,  2n  terms,  2n  +  1  terms,  of  the  series: 

1.  The  natural  numbers  ;  the  odd  numbers  ;  the  even  numbers^ 

2.  The  numbers  of  the  form  r  -^-hx  wherein  r,  k  are  constant 

integers  and  x  a  variable  integer. 

3.  The  distances  passed  over  in  successive  seconds  by  a  falling 

body,  starting  from  rest  (16.1,  48.3,  80.5,  •••  feet,  or 
4.9,  14.7,  24.5,  •••  meters). 

4.  1,  -2,  +3,  -4,  ...  ;  1,  -3,  +5,  "7,  +9,  ...  ;  3,  2f,  24,  .... 

5.  One  hundred  stones  are  placed  in  a  line  on  the  ground  a 

meter  apart,  and  a  basket  is  placed  a  meter  from  the  first 
stone  ;  how  many  kilometers  must  a  man  run,  who,  start- 
ing from  the  basket,  picks  up  all  the  stones,  one  by  one, 
and  returns  to  the  basket  each  time  he  picks  up  a  stone  ? 
...  8.    Find  the  five  elements  of  the  arithmetic  progressions  : 

6.  1,  3,  5,  ...  99;  1,  3,  5,  ...  2  A: -1  ;  4  +  5  +  6  +  ...  =  5350. 

7.  5. ..7 means... 75  ;  3...  11  means 11  ;  2^...3  means  ...  20. 

8.  ...  5  terms  ...  19  ...  7  means  ...  67 ;  1,  a;,  ...  4a;,  19  ; 

1^_...  4-50  =  204; 

9.  Fill  out  the  arithmetic  progressions  : 

04 ^-3..._|-4=:  10,  =  18,  =2(4A;-f  1).  [A;  any  integer 

10.  Find  the  distances  passed  over  by  a  body  falling  from  rest 

in  successive  quarter  seconds ;  and  in  successive  periods 
of  5  seconds.  [ex.  3 

11.  A  stone  thrown  into  the  air  took  5  seconds  to  rise  and  fall 

to  the  same  level ;  how  high  was  it  thrown?  [ex.  3 

12.  Find  the  condition  that  a,  6,  c  may  be  the  pth,  gth,  rth 

terms  of  an  arithmetic  progression ;  if  this  condition  be 
satisfied,  and  if  a,  6,  c  be  positive  integers,  show  that 
p,  g,  r  may  be  the  ath,  5th,  cth  terms  of  an  arithmetic 
progression,  and  that  the  product  of  the  common  differ- 
ences of  the  two  progressions  is  unity. 


406  SERIES.  [XII. 

13.  Divide  unity  into  4  parts  in  arithmetic  progression,  such 

that  the  sum  of  their  cubes  shall  be  ■^^. 

14.  The  interior  angles  of  a  rectilinear  figure  are  in  arithmetic 

progression  ;  the  least  angle  is  120°  and  the  common 
difference  5°  ;  find  the  number  of  sides. 

15.  A  three-digit  number  is  26  times  the  sum  of   its  digits ; 

the  digits  are  in  arithmetic  progression ;  if  396  be  added 
to  the  number,  the  digits  are  reversed :  find  the  number. 

16.  At  4  P.M.,  A,  riding  4  miles  an  hour,  is  11  miles  ahead  of  b  ; 

B  increases  his  speed  regularh^  :|^  of  a  mile  every  hour, 
and  has  ridden  since  starting  at  11  p.m.  the  day  before, 
72^^  miles ;  when  did  a  pass  b,  and  when  will  b  pass  a? 

§2. 

...  19.    Find  the  last  term,  and  the  sum,  of  10  terms,  n  terms, 
oc  terms,  of  the  series : 

17.  The  integral  powers  of  ±2;  ±3;  ±k;  ±i;  ±i;  ±i.  [k^l 

19.  l+()  +  ()-|  +  ...;   .672672...;  |  +  |+A  +  A  +  .... 

20.  A  man  invests  $100  half-yearly  in  stocks  that  pay  3  per 

cent  half-yearly  dividends,  and  invests  the  dividends  as 
they  are  received ;  how  much  will  he  have  invested  at 
the  end  of  10,  20,  30  years? 

21.  A  man  at  20  insures  his  life  for  $2000,  paying  therefor  a 

premium  of  $20  half-yearly ;  what  is  the  gain  or  loss  to 
the  insurance  company  if  he  die  at  30,  40,  50,  60,  70, 
estimating  that  it  costs  the  compan^^  10  per  cent  of  its 
premiums  to  collect  and  care  for  them,  and  that  money 
is  worth  5  per  cent  per  annum  ? 

22.  Show  that  V-444...  =  . 666-..;  -^2.370370...  =  1.333.... 

23.  Find  four  geometric  means  between  1  and  32  ;  two  between 

.1  and  100  ;  three  between  ^  and  9  ;  three  between  2  and  J. 


§14.]  EXA^IPLES.  407 

24.  The  sum  of  three  numbers  in  geometric  progression  is  13, 

and  the  product  of  the  mean  and  the  sum  of  the  extremes 
is  30  ;  what  are  the  numbers  ? 

25.  Show  that,  if  n  geometric  means  lie  between  a  and  c,  their 

n 

product  is  {acy. 

26.  If  the  common  ratio  of  a  geometric  progression  be  less  than 

^,  prove  that  every  term  is  greater  than  the  sum  of  all 
the  terms  that  follow  it. 

27.  What  is  the  condition  that  a,,  b,  c  may  be  the  pth,  gth,  rih 

terms  of  a  geometric  progression  ?  If  this  condition  be 
satisfied,  and  log^a,  log^6,  log^c  be  positive  whole  num- 
bers a',  b\  c',  show  that  a^,'a«,  a*"  are  the  a'th,  b'th,  c'th 
terms  of  a  geometric  progression. 

28.  If  there  be  an  infinite  number  of  infinite  decreasing  geo- 

metric progressions,  wherein  the  ratio  is  common,  and 
the  first  term  of  each  is  the  nth.  term  of  that  just  before 
it,  show  that  their  sum  is  a  :  (1  —  r)  (1  —  r''"^) . 

29.  There  are  two  infinite  decreasing  geometric  progressions, 

each  beginning  with  1,  whose  sums  are  s,  s' :  prove  that 
the  sum  of  the  series  formed  by  multiplying  their  corre- 
sponding terms  is  ss' :  (s  4-  s' — 1). 

§3. 

30.  Continue  in  both  directions  the  harmonic  progressions : 

2,3,6;  3,  4,  6  ;  1,  1^,  If ;  to  five  terms,  to  n  terms. 

31.  The  difference  of  two  numbers  is  8  and  their  harmonic  mean 

is  1| ;  what  are  the  numbers? 

32.  What  is  the  condition  that  a,  &,  c  be  the  pth,  gth,  ?'th  terms 

of  a  harmonic  progression  ? 

33.  If  a,  6,  c,  ...  be  in  geometric  progression,  and  a^  =  b^  =  c'' 

=  ...,  thenp,  g,  r.  ...  are  in  harmonic  progression. 

34.  Prove  that  the  arithmetic,  geometric,  and  harmonic  means 

of  two  numbers  greater  than  unity  are  in  descending  order 
of  magnitude. 


408  SERIES.  [XII. 

§4. 
•  ••63.   Determine  which  of  the  series  are  convergent : 

35.  2  +  5  +  1  +  ...;  1+3^3^  +  ...;  _L  +  A+A+.... 

1^2^3^      '      ^2^22^       '    100^100^100^ 

36  1  ,  ^-t-p.3      m  +  2p    3^      m^-Sp   3^ 

1      *5  2       '5-  3       '53 

37  i  +  ?.l  +  5.i_  +  i  J_+       .     ct  +  /^   1      a  +  2^  J_ 

1*2      2*  2^  "^3*23  '     b-^k'r      b-h'2k'  r't'"' 

38.  -l.+  J_+J.  +  ...;    __L_  +  _L_+__L_+.... 
1-2     2.3     3.4^  V(l-2)      V(2-3)      V(3-4) 

39.  J_  +  J_+J-  +  ...;    i+?  +  i:_5  +  ?^^7         . 
1.3      2.4      3^5  '  2      2.3      2. 3^4 

40. H -\ +.-. 

a(a  +  6)       (a  +  6)  (a  +  2  6)       (a  +  2  6)  (a  +  3  6) 

41.  _^+_«_  +  _^+...;    X  +  ^+A.  +  .... 
1^2.3      2.3.4      3.4-5  2.3      3.4      4.5 

43.  1+1  +  1:1  +  1:1:5  +  ...;  Jl  +  g!:f +  2^-^^-6^  +  .... 

2      2.4      2.4.6  4!  6!  8! 

44.  Find  S5,  and  its  bounds  of  error,  in  each  of  the  above  series. 

45.  "Write  the  above  series  to  powers  of  x  so  that  x""  shall  have 

for  coefficient  the  7ith  term  of  the  series,  and  determine 
the  radius  of  convergence  in  each  case. 
§5. 

46.  i_2  +  3-4  +  ...;     1-1+1-1+.... 

47.  a  —  h-{-c-^a^b  —  c  —  a-\-b-\-c-\-a  —  b-\-c-\-"'. 

48.  2-5  +  1-5  +  -;     1-1  +  1-1  +  ...; 

23      4  2^0      4 

49     i_l4.1_l+..,.     2  +  1      2^+1      23+1   '    ,, 
357  2  22    ^2^ 

g-l      2(2a-l)      3(3a-l) 
•       P  22         ^        32 

51    J L  +  J •  J L  +  J 

'    1-2      2.3      3.4  '    1.3      2.4      3.5 


§  14.]  EXAMPLES.  409 

52     _2    _J_  +  JL 5_  1     1   I   1-3      1-3.5 

■   3-5      5-7      7-9      9-11  '  2      2-4      2-4.6         ' 

53.  l-i.  +  ll±-i!l^+.... 

2^      22.42      22.42.62 

§6. 

54.  Write  the  series  in  §  §  4,  5  to  rising  powers  of  o^,  a;  —  1 ,  a;  +  1, 

oj— 2,  a;  +  2,  ic— 1+^\  a?— 2  +  **?  so  that  aj",  (a?— 1)",  •••, 
shall  have  for  coefficient  the  wth  term  of  the  series,  and 
construct  the  circles  of  convergence  of  the  resulting 
series.  o  ^ 

55.  Determine  in  advance  from  the  character  of  the  functions 

in  Exs.  56-8,  G6,  67,  69-71,  what  will  be  the  radius  of 
convergence  of  their  expansions  to  rising  powers  of  x. 


56. 
57. 

58. 


58.  Expand  into  series  to  rising  powers  of  x,  the  fractions  : 
1  3aj  — 2  5  — 10a;     .  1 


3-2a;'    (a;-l)(a;-2)(a;-3)  '  2-a;-3a2     l-x-^oc^ 

X .         1-a^  1  1 

(l-a;)(l-aa;)  '    2-2a;-a;2'    (lH-a;)2'    (l+aj)^* 
11  1  a—x       a?—Q(? 


•  ••65.    Find  the  scale  and  sum  of  the  recurring  series  : 

59.  4  +  9a;  +  21x2  +  51ar^+...;    1 +  3  a;+ 2x2-aj3 . 

60.  l+3a;  +  lla;2  +  43ar^+...;    1  +  2a;  +  3a;2 +  4a;34- •-. 

61.  l+3a;4-6a^  +  10a^+15a;^  +  21a;^  +  ...; 

\-x^x''-x'-\-o?-:i^-\-'". 

62.  l  +  3a;+5a^+7a^+..-;    l+a;  +  2a^+2a:3H-3a;4+3a;5+- 

63.  l  +  l  +  ?  +  i  +  A  +  ...;    l  +  A^+i9.^2  +  _6L^+.... 

-r   ^4^2      16  6      36        216         1296 

(a+l)-a      (a+l)^-a%  ,   (a  +  l)^-o^^2  ,  ... 
*      a(a  +  l)    ^    a2(a+l)2      "^    d\a-\-\Y 

65.    l+2a:  +  5a^4-|«^  +  f|a^^  +  j^'+-;  4  +  3+|+^  +  - 
2  2  16  4  o      y 

•  ••  67.    Expand  into  series  to  rising  powers  of  a;,  the  surds : 


410  SERIES.  [xn. 

67  1         .  1  .  1  .  1 

68.   Find  the  values  correct  to  four  decimal  places  of: 

V3  ;  V^  ;  ^9  ;  ^31  ;  </17  ;  ^80  ;  ^33  ;  -^240  ;  -^'720. 

•  ••  71.   Resolve  into  sums  of  partial  fractions,  the  fractions : 

gg     3a;— 2 .       5  + 6a;    .     1  -\-4x-\-a^ 

{x-l){x-2){x-3)'     (l-3a;)2'        (l-x)* 

70  ^^        '  {(i  —  b)x  . s^ 

1  — 2a;-|-ar''    x^  —  {a -\- b)x -\- ab  ^    {x  —  a){x  —  b){x  —  c)' 

7j     a^  .     ar^-a;-3 .     2ar^-7a;H-l 

'   a^-A3^-\-5x-2'     a;(ar^-4)  '  a^+l 

•••  75.   Resolve  each  term  into  its  partial  fractions,  and  by  aid 
of  the  series  so  formed  find  the  sum  of  the  series : 

72.  ^  +  JL+...-^l^;A+A4-  ' 


1.2  2.3  71(71  +  1)  '  1.4      2.5  w(7i  +  3)' 

73.  JL-|-J_4,J_  +  ...;       4  5  6 

1.3  3.5      5.7  1.2.3      2.3.4      3.4.5 

74.  1        +        '        +...,   __L_  +  _A_  +  . 
1.2.3.4      2.3.4.5  1.3.5.7      3.5.7.9 

y  K     X o^ a-x 


(1  +  a;)  (1  +  ax)      (l+tia;)  (l  -f  d'x)      (1  +  a^x)  (1  +  a^a;) 

+  ...+ ?^ 

(l+a«-^a;)(l+a«a;) 

76.  Resolve  into  partial  fractions  the  fraction  : 

— ; — ; ,  and  show  that  when  p<n, 

{x-a^){x-a2)-"{x-a^) 

qr!: _^ al^ 

(tti— a2)(ai— a3)...(ai— a„)      (^2— «i)(a2— a3)--(«2— ««) 

-\ 1 ^ =0. 

(«n-ai)(a«-a2)"-(an-a„-i) 

•••81.    Revert  the  series  : 

77.  y  =  x^-{-x;   y  =  Ax  —  x^',   y=lSx  —  6a^. 

78.  y  =  6x'^-\-x;   y  =  Sx  —  xr',   y=x^  —  15x. 

79.  y=20x-\-10x^  —  x*;    y  =  x-^af  +  x^ -\-x* -j- -". 

80.  yz=ax  +  bx^;   y  =  ox-\-ho? -{-ex? ;  y=aa^-\-bx!^-\-ca^-\-da^. 

81.  2/  =  (a  — aj)~M    y  =  x{a  —  xy^(b  —  xy^',    y^  =  a  —  x^. 


§  14.]  EXAMPLES.  411 

82.  Show  that,  ity=  ax""  +  bx^'+p  +  cx''+^p  +  •  •  • , 

then         X  =  A2/^  +  B?/^+^  +  C2/^+2^  +  •  •  • , 
wherein  n  =  1 :  n,     p  =  Np,     a  =  a~^,     b  =  —  na"+^+^6  ; 
show  that  n,p,  a,  •••  are  the  same  functions  of  N,  p,  a,  ••• 
as  N,  p,  A,  •••  are  of  ti,  j),  a,  •••. 

§  9- 

•  ••87.   Expand  to  six  terms,  and  write  the  general  term  of : 

83.  ^{1-x);    -^{l-x)',    -^(1-af)',  -^(l+x)-,  i/(l-px). 

84.  {l-xyi;    (l-x)-^',    (l  +  a;)"*;    {l-x)-'-,    (l-x)-K 

85.  (1+0^)-^    (1-ary^;    (a'-x^y^;   (i  +  a; +  a^+ •••)"• 

^^-  ^^  '    (l_3ar)^'    in^-'   ^'+^^   '    (1+^-y 

87.  (H-3a;  +  5ar^+...)^;    (1  +  2a;  + 3  a.-2  +  4a^+ •••)"• 

88.  Find  the  radius  of  convergence  of  the  series  in  Exs.  56-87. 

89.  By  aid  of  the  binomial  theorem  compute  the  values,  correct 

to  5  decimal  places,  of  the  surds  in  Ex.  68. 

§10. 

•  ••  92.    Find  the  sum  of  tlie  first  5  terms,  20  terms,  7i  terms  of : 

90.  1+2+3  +  -,   P+22+32+...,   13+23+3^+.... 

91.  1+3  +  5  +  ...,     12+32_|.52_|_...^     I3_|_33^53_f_..., 


92.    a  +  a  +  d  +  a  +  2d  +  ...,  o^+a +  d^  + a  + 2c?  +  •••, 


■i3 


93.  Find  the  series  of  values  that  x^—  5a^+  4:X^—  3ic  —  8  takes 

when  x=  1,  2,  3,  •••,  and  plat  the  function. 
Find  the  sum  of  5  terms,  20  terms,  n  terms  of  this  series. 

94.  Find  an  entire  function  of  x  that  shall  take  the  respective 

values  4,  6,  10,  when  x=l,  2,  3. 

95.  Find  the  5th  term,  20th  term,  nth  term  of  the  series  of 

figurate  numbers  :  1,  1,  ••• ;  1,  2,  3,  •••  ;  1,  3,  6, 10,  •••  ; 
1,  4, 10,  20,  35,  •••  ;  1,  5,  15,  35,  70,  126,  •••. 

96.  If  shot  be  piled  in  a  triangular  pyramid,  find  how  many 

shot  there  are  in  the  5th,  20th,  nth  courses,  counting 
down  from  the  top ;  and  how  many  shot  altogether  in 
the  5  upper  courses,  20  courses,  n  courses. 

97.  So,  if  piled  in  a  square  pyramid. 


412  SERIES.  [xn.  §  14. 

98.  So,  if  piled  in  rectangular  form,  with  p  shot  more  in  the 

length  than  in  the  breadth  of  any  course. 

99.  So,  if  the  piles  be  incomplete,  with  2  courses,  12  courses, 

m  courses  gone  from  the  top. 

100.  Show  that  the  sura  of  the  cubes  of  the  first  n  natural 

numbers  is  the  square  of  the  sum  of  the  numbers. 

§n. 

101.  From  the  tables  take  out  the  logarithms  of  500,  510,  520, 

530,  and  interpolate  the  logarithms  of  503,  509,  521. 
Test  the  work  by  comparing  the  results  with  the  tables. 

102.  Given   V3  =  1-732,   V5  =  2.236,    V7  =  2.646,   V9  =  3, 

Vll=  3.317;  interpolate  -^2,  V^^  V^'  V^»  V^^- 

103.  Given  the  squares  of  1,3,  5,  7,  9,  11;    interpolate  the 

squares  of  2,  4,  6,  8,  10. 

104.  Given  the  amount  of  one  dollar  at  compound  interest: 

for  1  3'ear,  1.06;  for  2  years,  1.1236;  for  3  years, 
1.19102;  for  4  years,  1.26248;  for  5  years,  1.33823; 
interpolate  the  amounts  for  ^,  IJ,  2|-,  3^,  4^  years. 

§12. 

105.  Prove  the  equation  : 

logea;  =  i[log«  (a;+  l)  +  log,(a;-  1)] 

+  (2x2-l)-i  +  i(2a;^-l)-^  +  i(2:c2-l)-^  +  .... 

106.  Assuming  the  expansion  of  log^(l  ■\-x)  and  of  e*,  show  that 

{\ -\- n~^ xY  =  e""  {1  —  ^n~^ 7?  ) ,    when   n  =  co. 

1 07.  Find  the  coeflScieut  of  a;"  in  the  expansion  of  -— — — — . 

108.  Expand  to  rising  powers  of  x ;  also,  to  falling  powers : 

log(a  +  &aJ  +  ca^),    \og\_{a? -\- px  +  q)  :  {a^ -\-p'x-\-q')']. 

109.  Expand  to  five  terms  by  Maclaurin's  theorem  : 

a;(e*_i)-i,    {a  +  bx  +  c^y. 

§13. 

110.  Compute  a  table  of  Napierian  and  of  common  logarithms, 

each  correct  to  four  places,  of  the  numbers  from  1  to  20. 


.z.'^i^,'":?'^^ 


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